Yes, I do. I had lenses with cylinders (for correcting it) in the past and didn't like them. They're larger and asymmetric - the lower side must be heavier so that they always align correctly on your eye.
With not-so-bad astigmatism, you can get away with not correcting it in your contact lenses. That's what I currently do. It works okay and brings you to a good level of vision (1.0 here in Germany, probably 20/20 in the US).
Ahhh, that won't work for me. I started off like that, but getting cylinders made a huge difference in my contacts. I've tried the bottom-heavy ones, as well as the ones with a "1" that's supposed to be positioned at the bottom manually. Neither has worked too well.
The shifting position isn't a problem, I think the bottom-heavy ones adjust fine with a little bit of practice, but the experience is never that good.
I have astigmatism as well and the ones I use are acuvue daily moist. They apparently use some blinking technology to keep them stable. To be honest, I didn't know they were supposed to remain stable until I saw your comments, so I guess they work well for me. There are only very specific situations where I get discomfort and they misalign for me, but a few blinks puts them back in their spot.