I can't tell you about UTSW Austin because I've never been there, but I did my medicine internship and neurology residency at the UW. It's a great place to train. I saw everything imaginable and felt very comfortable when I started practice from day one. Actually, practice for me in the beginning was a bit anticlimatic because not everybody I saw in real life was airlifted in from one of the neighboring five states in status epilepticus or myasthenic crisis. So, for the first couple months I'd look at my patient and go, "Really? You just have a migraine? Are you sure you didn't dissect a vert somewhere back there?" If you can survive residency (at least at the UW) you can handle all basic neuro cases (I mean that's the whole point of residency, isn't it?)
Residency itself wherever you do it is very much a sink-or-swim experience. When you're on call, you are the neurologist. But back-up from your seniors and attendings are always available at the UW and they want you to call if you feel uncomfortable at anytime. In fact, in certain situations you have to call, like when you need to push t-PA. I can't tell you much about research because I'm in clinical practice, but opportunities for research abound. Seattle is a very laid back place to train. You can actually have a real conversation with the surgeons! A much welcoming change from my East Coast med school. Work load is higher than when you're a med student, but that's anywhere you go. If you end up training at a place where there are more residents than patients, I wouldn't trust you when you're an attending. Think about it. Overall, I couldn't have imagined training anywhere else, and everything I've learned in residency is still paying off in spades everyday in practice. Good luck!