From the photo, that bike is probably a 1994 model with an upgraded (1996-97) XT rear derailleur. Definitely not '89-90.

If you know the problem is with the shifters, I would suggest trying to find original(-ish) replacements on eBay rather than buying 'modern' replacements, for the simple reason that the original shifters were quite good quality, while 7 speed replacements today will be bottom of the barrel quality (Deore LX vs Tourney) as drivetrains have gained more and more speeds.

These appear to be the exact brake/shifter combo your bike has, and are not too expensive: Vintage MTB Mountain Bike Shimano Deore LX M560 Shifters Levers | eBay

-Robin