Fwiw, i have the same bike - 1990 Trek 1400.

Over a decade back, I too had BB spindle play. IIRC, I replaced the ball bearings and repacked it with new grease (Phil Wood). That solved the problem nicely and it has worked flawlessly ever since.

Earlier this year, in a (not really uncommon) moment of OCD weakness, I replaced the cranks to a later model 170mm Shimano 105 (so that all my road bikes have the same crank arm length) and used the appropriate Shimano (Octalink) cartridge BB.



However, its gone through other upgrades over the years as well, including a threadless carbon fork, and STI brifters. Here she is in her latest iteration:



I was primarily a MTBer in my former life when I first got her, and when I rediscovered cycling and embraced being a roadie, I had bought a new primary road bike, so my Trek only has single digit thousands on the odometer.