1998 Jeep Cherokee BAD SHAKE IN STEERING - JEEP

check for looseness in the tie rods and other front end steering components

if those are all good i would replace the steering dampner.

Don't overlook the shocks. They're job is to hold the tire on the ground after hitting bumps. If ONLY one shock on the front is BAD, That tire will try to bounce more than the other one and feel like a shake in the steering because you feel it in the steering wheel. Put a knee on the bumper and push down real hard and see how many bounces it has after you do that. It should come back up and stop on the way back down. If it does more, consider replacing the shocks.

This sounds like this may be a case of 'death wobble' as it's referred to by some members of the Jeep community. It's usually caused by a worn component in the front suspension. I would recommend having the front suspension and steering components looked at, and have the steering stabilizer replaced.

The shocks and steering stabilizer can be relatively inexpensive and easy to replace. Although they're not typically the primary cause of the wobble, if they've failed they can be a contributing factor.