^^^A good southern car is a great idea. You can get them for the same price as a northern car, and they have no rust. You should undercoat it to protect it from the salt.
If you hotrod it a little, the kids might like it better. Kids love mine, and it rumbles a little. A paint job with some flames, and installing Fleetwood rear vanities would appeal to their vanity (you could only do this with a Caprice). You could even put Impala springs in it for about 1 inch of lowering (and an alignment). Some slightly bigger wheels would appeal to the kids too (all of them including you). Some video screens for the back seats might help them like the car as well. If you could make the video screens fit in the Fleetwood rear vanities, they would flip up into the headliner.
Your current pulling may be the tires. Swap the front tires side to side, and see if the pull follows the tire. You could put the front that it pulls to, on the rear, and rear on the front as an option.