Advice on mathematical thinking and problem-solving

I don't know that this is an adequate answer for the question, but I certainly have sympathy for the trap of getting caught up in details ... that may be very subordinate. For myself, although I've mostly managed to avoid getting stuck in details, when I was much younger I would occasionally-and-unfortunately fixate on small things, since I'd been led to believe that every detail had the same significance in mathematics.

The latter is only "formally true", in the idealistic sense that if any link in a chain of logical reasoning fails, then the whole fails. However, live mathematics is not so "boolean" in its legitimacy or art. In particular, NOT all details are of equal significance in mathematical real life, despite various logical ideals.

I do also tell my PhD students and other grad students this, that one should be willing to let quite a few details be postponed, and try to discern the significant ones... all the more so because many of the small details become completely clear (only) with sufficient hindsight. Truly, in a strong sense, many details are genuinely unfathomable "in prospect", since the true explanation will only come later. That is, even if one does want to insist on careful explanations, the immediate formal seeming-explanations in typical sources are in fact not correct... and therefore all the more unpersuasive or baffling. Thus, duh, leading a serious person to be baffled, and think that it's their own internal "problem", rather than appreciating (since we are not often let in on the secret) that purely logical correctness is not at all a reliable explanation.

Especially if one is sensitive to such things, the disconnect can be nearly fatal, or at least severely impairing.

I hope there will be other answers about other aspects...

Have you ever noticed that after the exam is over, on your way back home you quickly figure out that math problem without even trying all that hard?

During the exam, when your work goes around in a loop, your mind needs a break. You need to allow your mind to wander off, to give it a chance to end up somewhere outside of that loop.

Mathematics is also famous for different ways of attacking the same problem. Sometimes, some approaches are deliberately long and elaborated in order to keep the topic within the boundary of a subject (algebra, analysis, geometry etc.). Luckily (or when we are lucky), those approaches have alternatives which some people find more intuitive. Of course knowing different approaches to a problem tends to be very useful too.

Let me give you a few personal examples...

• I used to have troubles with understanding this proof of Fermat's Two Squares Theorem, while Minkowski's approach is so much easier.
• I only understood properly the $\varepsilon - \delta $ techniques in analysis (that was ~25 years ago) after my teacher plotted an alternative model on the $x$ axis and roughly defined what a vicinity is. A few years later, I discovered topology as a new subject in mathematics.

To conclude, try and seek for alternatives. Ask, if you can't find any (including on MSE). Maybe you are more drawn towards geometrical understanding, rather than "dry" $\varepsilon - \delta $ (as an example)? Once you develop a good intuition, you can attack long and elaborated methods.