Since you mention that you didn't notice the wobble as this pronounced prior to trying it on these two sets of wheels, my guess is the retaining ring/outer bearing race is possibly loose. In my experience with Shimano 600 and DA 6 speed freewheels is that the plastic seal cover (what you see the printing on) and the retaining ring are basically hand tight.

The red Park two in spanner fits the holes on top of both. Remember this is threaded in reverse. First remove the plastic sealing ring. Then see how much play is between the outer and inner bodies by pushing them up and down. Basically there should be none yet the bodies rotate smoothly with only friction from the grease. If there is play, tighten until the play is gone and the movement is smooth. Install the plastic seal and tighten as hard as you can without damaging the holes.

There's a chance the smallest sprockets are worn. If you have two chain whips try to unthread the smallest sprocket The remainder slip on and off and are held by splines. Take pictures of the spoke facing side of the sprocket and post here for examination. BTW, I have a great supply of silver 6 speed 600 & DA small sprockets. PM me if you need further assistance.