im not sure what you mean by a mind of its own but had a wandering problem on my car a little while back. if i accelerated it would steer to the right but id be holding the steering wheel straight. lift off and it would steer back straight.

the problem turned out to be crappy flo flex front lower arm bushes. they were split and soft so was allowing the arm to pull forward under accelaration causin a big toe in on one side of the car.

changed them and problem solved