The difficult part for me is that I have to force myself to ask the questions that you talk about, it's definitely not intuitive for me and my first reaction is not to think about the different solutions a problem could have, like you said I can't break a problem into chunks, and this I think is essential to be a programmer.
I read somewhere above that you spend a lot of time fixing bugs. Congratulations! You're already a programmer!
To move onward and upward, the biggest thing you can do is change your mindset from "how do I make this problem go away?" to "what is causing this problem to happen?" At the end of the day, you'll still get those bugs fixed, but you'll have a deeper understanding of the things that led to them in the first place.
Once you know that, breaking things down and shuffling ideas becomes second nature.