Canon 5D Mark II does full HD (1080p) movies. But its not a 'camcorder' and requires actually planning out your shots. Not sure if it has auto-focus in movie move. I believe a full episode of 'House' tv show was filmed using that camera a few years back. I have one - its an excellent camera - but I never use movie recording mode. You can get one used for 1300 or so and brand new for 1700 - 1800 - as its an older model.
If you buy the 60D now, you can get a new 60D from BHPhotovideo for $799, it's $999 with $200 off and I think free shipping.
The 7D also allows you to shoot videos, but that's probably out of your price range at $1199. It's the one that I've bought and am looking forward to using.
The 60D is a high end hobbyist camera. The controls aren't as good as the 7D, but the optics are fine and it's probably better to spend the extra money on better lenses. But, it's probably a good enough body that you'll be using it for years.
The 60D may fit your bill pretty well, it's a semi-professional model and it will be a vast improvement from your 350D. What I wanted to mention however is that it would not surprise me at all if Canon adds a 70D and a 7D Mk II to their line up in 2013, I'm not saying what and buy one of those but if you wait for them to come out you could get an even better price on the 7D or 60D. The T3i/T4i are viable options to look into as well!
I have no problem waiting for a better camera. At the moment, i'm in no rush to spend the money since things are tight. If necessary, I may have to relinquish part, or all, of the money to help pay bills. However, I definitely want to consider a new camera. Or perhaps I will consider After Effects, it's really up in the air. Mainly mention the camera since my 350D camera has served me well, but the DSLR is showing its age.