Gratulations first to realize where you f*** up in your studies and what you need to fix. Many do not realize it - make sure to tell everyone you ever met not to slack off in his studies. It is likely the most important lesson they can learn.
One thing that you seem to be stuck that makes no logical sense is this (and it may well solve the issue):
The vibe I get from my workplace is that they value self-learning over mentorship.
It is not self-learning OR mentorship. A mentorship is not there to TEACH you - it is there to GUIDE you. Otherwise it would be called a teacher, not a mentor. Optimally, you should have a team lead or something that will act as your mentor, and that includes working with you on directing your self study, realizing where you are weak, making sure you can participate with your strong points WHILE learning on your weak points, and answering the questions that you get stuck in your self-learning program. You may add some coursed together with self-learning (which I take more as RTFM - something that is always a core skill and that so many developers lack), but Mentoring is planning and helping you with your development, it is NOT teaching you.
And that is why junior developers should be assigned a senior as mentor-partner or a team lead should be there.
On the self learning side - your employers is not there to teach you thinks on their time that you should know and if they hire you based on your certifications then being unable to fulfill a job you should do is not "spend half a day learning" - it is "you are fired, we pay someone who did not slack off". Now, being a Junior developer with little experience (yes, that is what you are, despite internships - those generally are not worth a lot) gives you a lot of slack for some time, but you are expected to perform on the level appropriate for your shown time.
You obviously can not as per your own words. So, say welcome to 12 hour days 6-7 days a week. THAT is the price you pay for slacking. Say hello to very little holiday. You work in your contract, then you go home, grab a book and learn all the stuff you should have learned. Find a mentor, plan what to learn, but getting yourself into shape is not something any company will pay for unless they hired you for this exact purpose. So, you need to "go to the gym and get yourself in shape" - except that is not loosing weight, that is learning. On your own time.
You biggest problem is this: "I'm scared/worried about asking too many questions at work." - you need to get over that, find someone you can work with and work on a plan. Given that you lack skills as per your own word, here is a way how to deal with that: In pretty much every project team I ever was we had a guy who could not really pull his weight on anything complex. We never fired those we had because while they lacked skills, they totally made up on that with work ethics and willingness to do the crap work. Every project has boring work that still needs to be done. Non-exciting, grunt work. Your lack of skills count little when the goal is to write test cases to get the test coverage from 78% to 85% as mandated and most of those cases are boring as hell. Or when it is about fixing the styles on an internal web app because someone decided to change the look. Or keeping certain documentation up. THIS IS YOUR SAVING - if you become the go to guy for all that work, that in many teams means they will overlook your lack of skills because you ARE USEFUL. Now, this is obviously a "dead position" - no promotion here. But you do not look for that - what you look for is a place where you can contribute WILL LEARNING YOUR ASS OFF. Basically: Carry SOME weight so you are not dead weight, WHILE LEARNING.
And here is the bad news: No One will give you years to learn. You need to catch up, within half a year max. Now, good thing here is that what you need to learn is way thinner than your studies (not all is used on every project), but you must learn, fast. Goodbye to free time.
And learn how to learn from books. All your approaches speak "wasted time".
I attend 3 or so webinars a week on topics I want to be more familiar with.
Webinars are totally useless outside of giving you an overview of a topic. Well, mostly - some may demonstrate some arcane element, But in general they are TOO SMALL for being sensible.
I retain knowledge better when it's a hard lesson, so I'd rather spend 2 hours figuring
it out than bother someone who might already know.
There is a time for that, there is a time for asking. Here is the problem: if you take 2 hours figuring it out, I pay for 2 hours instead of 10 minutes. Not good.
I take 2 or 3 certifications a year, and the way I study for those varies.
Those are again mostly useless. I have zero certifications and run circles around anyone I ever met with - they can not really test in depth, they are mostly "book learning". THAT SAID: they at least demonstrate this, and that has a serious value in itself.
I spent a lot > of time on StackExchange reading questions about my technology and
formulating my own answers (I wasn't fast enough to be the first or best answerer on
many of them), practicing that way since I didn't have those problems in my work
I am not sure how much you get from that, given that most questions on Stackexchange in particular are QUITE low hanging fruits. It is a nice exercise on something, it is not a good approach on learning.
I found a few podcasts in my area and listened to them weekly.
That is, again, quite on the level of webinars - HARDLY worth the time. You just do not have the time to go into details.
I joined 3 local groups that have monthly meetings (outside of work hours).
Hate to say it, but you are on webinar territory again with even MORE overhead. If it is like any of the groups I ever joined, it is more for social gain than anything really in depth.
You know what is missing? READING BOOKS. Not blogs, books. Because a blog post is 4-5 book pages, and a book may be 400-600 pages on a specific topic, going in detail. Back when I was in school I was reading 500 page programming books on a weekend and developed a technique to memorize the table of content - I got the grasp, and knew what a book covered and roughly where any topic was and I trained looking things up FAST. And right behind me and in the living room are around 7 meters of books for relevant topics that I keep there as reference material. THAT is in depth knowledge, not some "90 minutes talk in a webinar". The moment you realize that RTFM and RTFB (read the f**** book) are where you really can gain insight, not by meeting people for a drink, is when you start making progress on getting in depth knowledge.
And if you think that is hard - that is how you become good and that is totally normal for medicine or law. And yes, I love blogs, I listen to webinars - but they are all "noise", checking interesting areas, getting a snippet here or there, NOT "learning structured".
I feel demotivated by my company's product and culture. It feels corporate and it's not
like they're changing the world or anything.
First, that is YOUR problem for two reasons. One, you choose your company, second - you have expectations that are not realistic for 99% of the population. Now, the choose part is this: there are different areas of making software and if you know what you find interesting, why you do not look for a job that - meets this? The pay may be worse (as in: A TON of people want to make computer games - guess what, games programmers are the bottom of the barrel regarding pay for that one reason, most of them) and you may have to do real grunt work for some time (particularly because you wasted doing internships in interesting ares - wasted chance) because besides being a bad programmer most interesting work will require a lot of domain knowledge that you lack, but...
... in general: you will not have an exciting career, you will work for someone doing something that does not excite you because that is what 99% of the people do.
If you are REALLY lucky, you may have a talent that matches up with something that excites you AND you may find that this also does pay you well (i.e. not "making games"), but otherwise you will choose a job that pays as good as you can and work for money. Do you think any of the accountants on the planet are excited about their work?
I strongly suggest you adjust expectations and realize that "earning money" is an AMAZING motivator. It allows you to have a family and have children wasting their time in university WITHOUT wondering how to pay all those bills.
If you are one of the really few luckies, you may find that database work is something you really enjoy (VERY few real db specialists around, so - somehow the pay is QUITE good), or some other field. if you are EXTREMELY lucky you find you have a real hand for something that makes money, and the energy to push through for a job that pays a TON of money in IT. Yeah, those mythical jobs exist, but - well, rare is rare.
But your really need to adjust the attitude here. Or your job. I have i.e. always worked freelance - but then this requires a deep understanding of a specific area or you may not get a decent rate. And even then - I focused on building things to be able to choose my customers because large companies (which make large projects and pay accordingly for specialists) ALWAYS have a ridiculous amount of "corporate". This is how the game works - sadly. But guess what - startups generally lack the money to pay high, so, unless you go that way... or win the lottery - the corporate feel is there to stay, sorry.