One way to sell a car is to call around to mechanics. Even good mechanics have clients whose cars are unrepairable or on the verge of collapse.
That’s how I bought my current car. My old car had a massive mechanical failure and I was talking to my mechanic about it when a guy called and asked if he sold cars. My mechanic said “no… but what kind of car is it?” He brought it over, I had it inspected and paid cash the same day.
The other places I looked for cars were in the newspaper and word of mouth (friends-of-friends).
The hardest thing about shopping for a car when you don’t have a car is getting around to look at them (people have this habit of living all over). I begged rides from friends for car-looking and kept on the DL for everything else. If you can afford to buy a new car without selling your old one, it’s much easier to deal with. I would recommend doing it that way if you can.
If your car is a Japanese model 100 -150K miles in reasonable repair selling around book value, and you live in a college town, I would expect to sell it in 2-3 days out of the newspaper (in Ann Arbor a Toyota with 150K for about $2k sells the day it is listed). If it is American with over 100K miles or has a major mechanical fault, it will take much longer.