I drove around the mountains of Idaho in winter for years in a FWD 2002 Toyota Corolla and never skidded once. $5k won't get you much in an AWD, I've seen very few old Subarus that haven't blown head gaskets by now. I don't know what the market is like where you live, but maybe keep an eye out for a 2000-2010ish Acura RL or TL with AWD. It'll have high miles at that price but they are good cars. The Rav4 as Jackofall mentioned is also great, or the Highlander but yeah bust out the $$$$$$
I'll also throw out the idea of an older AWD Ford Escape/Mazda Tribute. I had a 2006 that made it to 186k relatively problem free before I got hit and it was totaled. Safe car too, it saved my life, got hit at a full stop by someone texting and going 55mph.