Find another job. Quit.
4 years ago, I was in your boat. My company had downsized (second round of layoffs) and we were all asked to do more. Then a couple of (smart) people quit, and the workload on us basically doubled. The company itself was in trouble, especially after one of our main clients went bankrupt and a significant portion of the contract was not paid.
I was good for a while; I felt happy to contribute and try to help save the company. Slowly, it started happening. I didn't even notice it at first. I was always anxious. I had trouble falling asleep, trouble waking up, and when I did wake up I was more tired than when I went to sleep.
I was very tense, and it was harder to focus on work. The more I pushed myself, the harder it became to complete tasks. The thing was, I always loved programming and I always felt blessed to work on something I mostly enjoyed. This joy was sapped out of my job first, then out of everything I enjoyed in my life.
My wife was the first to comment on it; she said that I was becoming miserable and miserable to be around. Those were harsh words, and they stung, especially since I felt I was doing what was necessary for my company and my family. In retrospect, she was 100% right.
Then it happened. One day, I woke up, ate breakfast, got dressed and just stood at the door. Then I couldn't take another step. I just started bawling my eyes out. I mean full on sobbing. I sat on the floor and just let it all out. Fortunately my son had already left for school so he didn't see his father go through it, but my wife saw it all. I was so embarrassed but I couldn't stop. I ended up calling in sick that day.
My wife wanted me to completely be away from my computer and phone. Go for a walk, take a long bike ride, maybe see a doctor. I was a mess, but I kept checking my emails. And as the requests came in, I became even more tense.
I saw a doctor the second day. He said it was a job burnout like yours. He said I need to take a vacation or else I could never get better. I let my company know that I needed a few more days off and why, even though at the time I felt embarrassed and weak.
My manager sent a very nice reply but in the very last line, he reminded me of upcoming deadlines and milestones that had to be met.
That very same day, I got 3 or 4 emails requesting help on tasks or small updates to a client's site. Then the next day I got more requests. Finally, I told my manager that I needed a couple of weeks fully off just to be able to recover. My sleep situation alone was getting worse and worse. I felt guilty that I was letting my company down.
When they emailed and called me with more tasks the next day, I knew I could never get rest.
I ended up quitting my job a few days later, especially at my wife's insistence. It took me months to recover but I'm so happy that I did.
Along the way I learned a few things about myself and what had happened.
Stress is normal, and a part of most jobs. I actually felt like I excelled under pressure. However, constant and unending stress and tension especially on your mind adds up overtime, especially if you don't take a break. It's made worse when your job starts creeping into your life, like mine did.
There comes a very definite point where chronic stress becomes a burn out. You'll be getting through your job and life one day and the next day you won't be able to go on. When this point is reached and it seems to be that you're there, stop immediately. Don't push yourself further.
You WILL NOT recover until you take time off. Full timeoff. Your mind and body need a break. It's not a suggestion. Even if you intend to stay with the company, any short term losses will pall in comparison to you not being able to function.
This seems to hit more people in IT and software development than most job sectors (nurses as well). I think people underestimate how stressful a job like programming is "Oh, you're just typing away at a computer". The reality is, you have to deal with so many issues, and are likely writing apps that are being viewed by hundreds or thousands of people daily, and once a release is made, if it's broken you can't do anything about it until it's patched and re released. You may also be responsible for writing components that are financially critical and if a third party library or service acts up, you're the one responsible for fixing it. People don't care that service XYZ stopped working, they just care that your app is no longer functioning as it should.
Finally, just some thoughts. Please let this be a lesson in the future. Put hard limits on what you can do and what you're willing to do. Take breaks. Work for a company that respects IT and its employees. If you were so crucial to the company why is there no backup plan in case you're no longer there? People quit or die everyday. Also, if they were going to be putting themselves in such financial peril, why would they buy the competitor? This makes me feel like your company is very badly managed and put increased profits above its people. You are not responsible for its downfall.
Please take care of yourself. It took me a few months to recover but I was helped by the support from my family and friends. Along the way I've met a few people who went through similar circumstances and one thing I've noticed is that pushing on just makes things worse. People have gotten strokes, heart attacks or nervous breakdowns from pushing on. Please don't. Reading through your question just brought back everything I went through and I have so many regrets. Jobs and money come and go. You get one life. Your kids need you. They are dependent on you. You need balance in your life. Please consider everything that everyone has said. Not one person is suggesting you keep doing this. I've bookmarked this question, please update us and let us know how you're doing and what you end up doing.
Also, the reason I wrote find another job and the crossed it out as that you need time off. You also won't do well in interviews if you're still exhausted.