Well, you may have to look out of your area for an early 2000's Corolla(2003-09). Maybe you can find a good example that is also easy to fix but, it's very difficult to find any vehicle for a few thousand bucks. To me, anything that cheap is just cheap junk and already used up.
But, they're out there. You know, that creampuff(or cherry) that has low mileage for its age and well maintained, like having things replaced when broken and not left for the next owner to fix. However you must look hard & long and expand you viewing area while being willing to travel some.
You may even find a car that's so good you won't want to give it up!