Hi Becky!
For what my advice is worth, try the following.
Before fitting any Chuck or faceplate, or anything else, to the Mandrel, always ensure that the thread and register are clean. Don't forget the thread in the Chuck or backplate, also!
An old toothbrush is handy for cleaning threads, (or Taps) in this way.
1) Screw the Faceplate onto the Mandel, (it may be worth marking, after skimming, the Faceplate and the Mandrel – if possible, so that in the future you screw the Faceplate onto the same point each time), and then lightly skim the face of the plate until it cleans up all over.
2) Having done that, then lightly skim the Outer Diameter of the Faceplate, until that just cleans up all round.
Those operations should cure your Faceplate problems.
If the three jaw, or four jaw chucks wobble, I suggest that you remove the Backplate. If the backplate has been removed previously, there ought to be pop marks on Chuck Body and Backplate that are in alignment.
Are there? Or if they are away from each other, remove the Backplate, clean the plate and the register in the chuck body, and refit it with the marks aligned, and check if it still runs out or wobbles when fitted on to the Mandrel.
If the Chuck Body and Backplate have not been marked, BEFORE separating, pop mark each one with the marks close together, so that when refitting, they go back in the same relative position.
Screw the Backplate on to the Mandrel and again very lightly skim the face until it cleans up all over.
Do not skim the Outer Diameter, or you will lose the concentricity of the chuck.
Remove it from the Mandrel, wash or wipe it clean, and refit to the cleaned register in the Chuck Body.
Incidentally, do not expect a workpiece held in a three jaw chuck to run absolutely concentric. A good chuck will be within 0.005 inch, a bad/worn one may be anything up to 0.030 inch or more.
If there are diameters or bores that have to be concentric, machine them at the same time, without moving the workpiece.
If this is not possible, then for the second operation(s) the piece needs to be held in the four jaw chuck and the original diameter clocked to run true before starting to machine the later ones.
Sorry if this latter bit is what you already know, and do, not trying to teach you, or anyone else, to suck eggs!
Good Luck!
Howard