Hello Blake,
Spindle turning between centres would not create too much of a problem if centres are misaligned. Bowl turning from a faceplate to a chuck should not be a problem as the tailstock is not essential. The chances are there is some inaccuracy in your chuck or your tenon cutting. It is good practice to shape the outside of a bowl, cut the tenon and then true up the shape after mounting it in the chuck, before starting to turn out the inside. If you do want the extra support of the tailstock then it needs to be aligned.
The tailstock is the item which is moved quite regularly during turning. I have seen turners slip a piece of card under the tailstock to bring back the alignment but that is only ever a one off quick fix. Better to match the headstock to the tailstock. With a pointed centre of some sort in head and tail stock, adjust the headstock so that they meet. I think that your lathe has machine screws which fix the headstock to the bed which might provide the wiggle room needed if they are loosened.
Once aligned there should not be a problem. Good practice if possible is to bring up the tailstock centre with the piece of wood fixed to the faceplate or chuck with the lathe spinning.
ATB John