That will limit you some. In fact you might want to stick with what you have until it dies, and try and save up a bit more. You can buy a used car at that price, but it won't be much better than what you have. If you can get to about 8-10k, you can pick up a decent used jeep or small truck that will be good in the weather and won't have 100,000 plus miles.
I'd really look at the different types of jeeps, Cherokees, Wranglers, even the Liberty. Subarus are a bit pricier, at least used.
What is your commute? Hobbies? How much bad weather driving are you anticipating?
Probably won't be driving much, but basically 6 months of the year you're stuck in snow (this is Iowa). We saw an accord for about 4 grand. The big thing about mine is that the underside is hanging down and it sucks.