Read Visual Complex Analysis and do some of the exercises. And practice imagining the arguments and recalling them without looking. Best thing you could ever do for your intuition.
Having intuition is fine, but you shouldn't ignore "formalism". Work on all the skills needed to negotiate math and science. Intuition is only one of them.
Well, formalism is needed, too, but less so for an engineer. And you shouldn't take it for granted that people know what they are doing when they present things very formally. On the contrary, I've found it very productive to take it for granted that people usually present things too formally, and that there is a better way to understand things, which can be found by browsing around for the right book and thinking for yourself or talking to the right person, rather than taking what the books say at face value. It is usually not necessary to be so formal (actually, you can be as formal as you want, without raising my objections, as long as you ALSO explain informally what you are doing, if it's not immediately obvious). My experience has always proven that, up until I got to research level math, where I ran into trouble, but I don't think the OP is headed for math research. One of the big difficulties you come across if you are more intuitive is that you have to "swim against the stream", to quote V.I. Arnold's point of view on this issue, because most mathematicians these days are more formal/algebraic. So that artificially makes things more difficult for a more intuitive thinker. I'm not saying intuition is the only thing there is, but it's a kind of glue that holds things together, without which, you just end up forgetting things as soon as you learn them. That's why I think intuition is the most important thing, by far. Having pictures or gut feelings of how things work is what makes it stick in your mind (and for me is the main thing that gets me interested in the subject in the first place).
I think it's partly a matter of style, too. There's some leeway in how formal a mathematician you choose to be. In the past, very intuitive, non-rigorous people like Witten have had a big impact, and on the other hand, more formal people like Weierstrass have also made their mark (albeit more boringly).
If you are going to be a mathematician, it is true that you have to come to terms with the logic of doing proofs and writing things down formally at some point. It would be a shame to let the paranoia of being wrong overtake everything that's fun about math. You need the paranoia if--and only if--you are going to be proving serious theorems. If you're just an engineer, perhaps caution is enough, rather than paranoia.
