This sounds similar to a problem that I had in a similar project. I didn't think to increase the supsension strength, although my suspension code was pretty bad/unstable - setting the strength too high meant that my car wouldn't be able to stand still without randomly bouncing into infinity. I might try revisiting it and using that approach whenever I get back around to it.
The way I 'fixed' it in my game was by adding a secondary somewhat thin box collider underneath my real collider which had a different ice-like physics material that has slip set to 1 (max) and bounce set to 0. Should my suspension give out completely, the car->road contact then has 0 consequences