Is it runout or wobble? Runout is a constant thing. You rotate the lathe with the handwheel and everything is secure but the center wanders. the point closest to the rest is the same every time.
Wobble is not constant. When you rotate the lathe, the piece sort of flops with gravity.
Runout is usually the insert or bore in the chuck. It fits snugly against the nut or bearings on the lathe spindle, and if an insert, bottoms against the chuck body. If neither of these conditions can be met, it's swap-out time. You've got advice on that.
Wobble is probably a grip problem. Only thing that could permit it that's purely chuck-based would be a loose scroll. Chuck your piece with the help of your tailstock. Snug the chuck, remove the supporting point and rotate. Shouldn't wander.