The "chain" style alignment shops rarely have alignment techs that know how to align an older car properly. The factory specs for old cars are what they go by but those specs are not right for today's tires. I suspect that they didn't dial in enough CASTER. Caster provides stability by making the wheels want to keep the car going in a straight line.
Unfortunately, you have to be clear and tell them what you want instead of letting them align the car to factory specs.
Always ask for as much caster as possible. Some cars may not be able to get more than 2 degrees. Never settle for less than 1 degree or the feeling that you describe will surely be the result.
Maximum caster. 1/2 to 3/4 degree of Negative camber and 1/8" toe in.
My Charger has offset upper control arm bushings and was aligned to 5 degrees Positive caster left, 5.5 right. It tracks straight and is extremely stable at freeway speeds.
Good luck.