Over time many may have adjusted the steering that didn't always know much.You need to check for were the play is,and yes it can be in more then one place.
There maybe some sloopy parts that need replacing or rebuild.
How ever some times I find tire rods are off to one side vs both being even; If tow is right why would that matter? ; That can make your steeringbox run out of center and that in its self with make a good steeringbox feel bad on the road at speed.
So to start with make sure your steeringbox is on the center of its design when car wheels is strait forword. Almost all types of steeringboxs are designed with sloop on ether side of center but on center is very close to zero,is done that way so cars don't wonder going strait= if its not on center as it should be it feels funny an drives poorly,as so if some one trys to adjust it when the box is anyplace other then on center,they screw up the adjustment
What I'm saying is be sure its worked on by a reall front end man,there are very few of us left!!!!!