Dana 60's get the wobble from worn king pins and rod ends the steering stabilizers are needed to stop these. The steering stabilizer is probably original and worn. I would buy one of these before dumping money into other things. They are 31.00 and you need one if you have the wobbles.
A steering stabilizer will only mask a problem that needs attention. All it is is a small bore shock absorber. It can't remove play. If the stabilizer "fixes" the problem- don't believe it. The instability problem is still there, waiting to jump on you, and Murphey is the holding it's leash.
The king pins generally don't wear although they might suffer from corrosion or abrasion if they aren't lubed. The bushings are sacrificial and they and the bushing springs are the wear items. Change the gaskets if present, or use silicone, and put in new grease fittings. I fact, consider changing all the grease fittings. Your counting on a 25 year old spring an ball to let grease in and keep crud out?
Worn shocks can allow uneven tire wear, wheel hop, brake grabbing on bumps and uneven terrain, yadda, yadda, yadda. If the shocks are OEM, replace them. If in doubt, replace them. The springs on these things are too stiff to rely on the bumper bounce trick.
Springs that are sagging can cause issues because they don't provide the intended resistance to vertical travel. If the spring bolts are not centered in the eyes the rubber bushings need to be replaced. Polyurethane one are best. Bushings that aren't centered change the front end geometry and it's never for the better. Sagging springs are effectively longer than OEM and the axle doesn't know where it should ride. The difference is made up in instability.
The allowable play in all the front end components is 1/16" If the rod ends, drag link etc have more, replace them.
Check for bearing and hub play by resting one hand on top of the tire and pushing the truck away with the other. Also, raise each front wheel, one at a time, and grasp the front and rear of the tire. Try to rock the tire and feel for play. The first will usually show bearing play, the second could be anything attached to the wheel (drag link, etc) so do this after you verify the 1/16" or less play or it won't narrow down the possible problem.
If your steering does not return to center after a turn without a conscious input there are other issues.
Improper wheel alignment (caster and camber etc) along with the tire pressure can all cause instability. These trucks had play in the steering when they were driven off the assembly line. They will always have more play than a modern vehicle. Try to find an old timer for the alignment. In any case, consider talking to the alignment guy and see if he can set you up for a very gentle bias to one side or the other, preferably to the left. That will have you in a very gentle turn to one side and the play won't be something you feel going down the road. Since most of us drive with the left hand braced on the door somewhere and the crown of the road needs to be climbed to the left, a left bias is very comfortable.
The front end of any vehicle is a pretty complex place where everything needs to be correct.
OLDCHEV4X4 is a hero with these front ends. He builds and races full size Chevys off road. If you get really stumped, shoot him a PM.
Lance