Knowing which transmission you have, might help us if the mount doesn't cure the shifter.
My 1990 Yj has the 4.2l & AX15 trans. the shifter does move slightly from side to side. You can also twist it slightly, & move it up & down.
I wish it had a regular old school Hurst shifter, (bolts to the side of the trans.) I know how to tighten them up so they have zero play.