Several possibilities come to mind (in order of cost):

Tyre pressure is important to these over tyred sedans,

Out of spec. front wheel toe-in, especially with worn tyres will affect
tracking,

Correct front wheel alignment is is critical so if the earlier
suggestions are moot this is the place to start.

Since you just acquired this car you ought to consider that it may have
been in a wreck and is bent, throwing the original alignment out of
spec. The alignment shop will tell you so, especially if their
adjustments cannot return it to specification.

You don't cite any looseness in the steering but something may be loose!

And don't discount the rear suspension, it may be loose. The alignment
shop should check it as well.

It's not going to fix itself so now is the time to take care of the
problem.

try getting the tracking done

I bought my 300 TDT in 1993 with 144,000 miles on it.  The steering
wandered all over the place, seemingly similar to what you describe.
There was excessive play in the wheel, and the steering box could not
be adjusted to remove the play.

I found a steering box at a wrecking yard for $75.  There was a lot of
cussing and struggle to change out the box, but I got it done and it
fixed the problem.  I now have 410,000 miles on the car, and the $75
steering box has not had to be even adjusted and still works great.

Paul Fretheim

Had this problem with my '90 420 SEL. It turned out to be a bad/loose tie
rod end.

John