Wow! Thank you everyone for all of your suggestions. This has been incredibly helpful. I've got a pretty good picture of what to look for now and I'm sure I'm going to be searching for a while to find someone who matches it ;) Leave the head alone, run if there are gadgets involved, honest person who can communicate well and is still learning. Trust my gut. Got it!
I guess I should have said a bit more about myself in the first post. I would describe myself as a good rider. I can sit just about anything and I have fairly good posture. The unfortunate thing is that I've been -how do I say this kindly- messed up by multiple trainers. I have about 12 years of weekly lessons in my history: with the first person anything I learned was from the horses, and the next person, it turns out, taught me a lot of bad things (see-sawing on the rains, etc. Bleh!). That's why I'm taking the utmost care to find a really, really good person to teach me now. :P
and had a terribly conformed horse with no work ethic
Haha. Don't tell my horse I said this, but that describes her pretty accurately. I wouldn't be taking lessons with her, though.