Did you drain the old gas when you first got the car? I got a bad batch of gas one time and it did this. I drained the gas and got fresh high octane Shell gasoline with a can of seafoam in it and changed my fuel filters and it cleared up the issue

Hesitation on first applying gas....sounds like a vacuum leak somewhere.  Get a can of carb cleaner and start the car and start spraying around the motor.  Your intake gaskets might be shot.  You might have a torn line somewhere near the carb/distributor.