In case you are unaware, the question of how much of our thinking is language-constrained has been extensively studied in linguistics and cognitive science. http://en.wikipedia.org/wiki/Sapir-Whorf_hypothesis is a good starting point. Anecdotal evidence is great, but you should also look into the studies.

My personal perspective: I am natively trilingual, as an Indian person. (someone else mentioned this as well.) My two Indian languages have fallen to near-complete disuse, to the point that I'm only borderline fluent in them. Nevertheless, growing up with two or more languages forces a child to seek common denominators, and thus think in terms of imagery.

This is definitely the case for me. As soon as I focused on the word "empathy" for a few seconds, I actually had two different images come to mind. These were both exemplars, as in, specific instances of people in my life showing empathy, one from recent memory and one from long ago. That's the trick with imagery -- it can often be crude, and only cover a special case of the concept, but it still does the job much better than words. You mention precision. Precision can in fact be a disadvantage, given the inherent ambiguity of human thought.

This is learnable. It gets harder as you age, but never impossible. To improve thought imagery, try this. Buy a ginormous whiteboard and cover one wall of your room/office with it. Draw everything. You might initially have to struggle with coming up with any sort of image. Your perfectionism, and your poor perception of your drawing skills, if you suffer from those, might hinder you. Try and lose your inhibitions. When you're at your computer, keep looking at the whiteboard once in a while. You want to get to the point where if there's a concept you're working on over a few days, every time you think of that concept, you should immediately be able to see the corresponding picture in your mind's eye. Also keep reams of paper around. Again, draw everything. I'll even start drawing on napkins if a thought comes into my head that I need to draw and I can't find paper. Hopefully, you will eventually be able to 'draw' effortlessly in your head.

Here's a solution for your "lookup table" problem. This is a technique that is used in teaching accents, but I have also used it in learning new languages. Pick a character. It could be a friend, or a character from a TV show, just anyone who speaks Spanish (preferably exclusively). When you're trying to speak Spanish, be that person. Imagine you're them, to the extent possible. It will make it a lot easier to push the English out of your head.

Finally, subvocalization (i.e, reciting sentences to yourself). Also (un)-learnable. Start gliding your eyes over the text rapidly. At first, you won't understand anything. Try the same sentence again. Force yourself not to say the words. If it's too hard, go fast enough that you can't possibly say the words. Keep doing that until you derive at least some meaning from the sentence. Start from there, and hopefully you'll get better slowly.

These are all things I've used to various degrees. Thinking in English is not a big problem for me, but more imagery helps, no matter what. YMMV, but hope this all helps!