What's wrong with my rear wheel? Wheel moves about 1/4" front to back when brake is fully applied, cassette seems a bit wobbly, some rubbing noise
I've seen this issue in the past and the culprit was the axle (not the skewer) in the hub had snapped.

Fortunately, the bike was still rideable to a certain extent, as the QR skewer was holding the two broken sections of axle in place. Note that even when the wheel was removed from the frame, it still wasn't immediately obvious that the axle had snapped, until the skewer was removed from the hub. However the whole wheel was replaced ASAP for safety reasons.

Due to the age of the wheel (15-20 years), it was more economically viable to replace the wheel than to rebuild the old wheel with a new hub.