sounds like the spindle is fine. Now you need to get that backing plate threaded on ALL the way so that it sits flush on the spindle register (flat part perpendicular to the threads, closest to the headstock) before doing anything else. Then, once you've done that, face the backplate and recut the chuck register. The chuck "registers" to the back plate in 2 ways. The end of the chuck, where the bolts go, sits flat on the outside portion of the backplate - this is the bit that reduces runout or wobble that increases with distance from the chuck. The inside lip of the chuck sits against the ledge sticking out from the backplate - this is the bit that makes the chuck run perfectly inline with the spindle. If it's off, then the tip of a point held in the chuck with scribe a circle, instead of rotating in place.
Most likely you'll need to recut both. The flat register is pretty easy, the inside register (the last one above) is trickier and takes a lot of care. Also make sure that the bolt holes or even the countersunk holes are not interfering with the chuck bolts as that can throw the chuck off too.
I had to do all of the above with the chuck that came with my 6 x 18, now it shows ~0.001in runout and no wobble. Before it was in the 0.01+ range. Also, even once you've made those changes, you can often lightly tap the chuck while it's loosely screwed to the backplate to get rid of the last couple thou runout (I got mine from 0.004 to 0.001 that way).
If you do all of that and it's not fixed (unlikely from what you've described so far), then the chuck may be shot, but that's the very last thing you should worry about.