I tend to dig in and spend a lot of time and resources trying to solve it myself...
That looks like a really good starting point to acquire "asking skill". It allows you to establish a few simple and safe rules to follow. Having rules will help lower the reluctance by turning what has been an "exceptional case" into a formalized, regular routine - like shaving or regular workout in a gym.
First, set yourself a reasonable / respectable time limit for a digging. An hour or two, for example.
Primary goal here is to lower a stress by building a confidence that you put substantial effort into attempting to find solution yourself. Another purpose it serves is preventing you from looking like a help vampire, like someone permanently asking questions: this time limit basically guarantees that you don't ask questions more than once per hour or two (that's what I meant when writing "respectable" limit).
Second, after the "time limit" is exceeded, stop digging and... no, don't ask others yet... write down a brief overview / summary of what you were able to discover so far.
Primary goal here, again, is to lower the stress, to cool it down by carefully analyzing, summarizing and writing down your efforts and findings. This also will likely help making further discussing the question smoother since you will build it upon a structured summary instead of chaotic bits from your "raw" research.
Third, review your original question in the light of the summary you just made and "polish" it if needed.
Main point here is to simply ensure that you covered everything you could within given rules.
- Somewhat tricky thing here is at this moment, you may get a feeling that improved question has an answer reachable in one minute of web search. You should follow that feeling, since suppressing it may increase stress - with the only reservation that it shouldn't led you into another hour of digging, which would in turn bring you into another round of preparing to ask others. Set yourself a "secondary time limit", really small one - 5 minutes for example. If the answer is not found, just "rollback" to where you were 5 minutes ago - summary of prior findings plus polished question.
Fourth step is, well, come to someone else and ask your (reviewed / polished) question. No need to explain what you have found or present your summary at this moment - but be prepared to quickly and clearly cover that if asked.
That's it. Just practice that routine, and the more you practice, the easier it will get.
I myself experienced the "transition" in both directions - first, from culture based on learning-from-others to one favoring autonomous-research and then back to asking-others. Have to admit, transition to asking-others went quite tough - maybe because it feels soo rewarding when you find the solution yourself. And the knowledge acquired by doing own research sticks much better in my experience. If only it would take not so much time...