Rocking could be a tweaked frame or bad axle bushings too. Re-tweaking is possible. Also may need to re-stake things if its bad.
Bad axle bushings can compound the gear issue, or be the issue alone. Grab a wheel and feel for an oblong bearing. An oblong bearing causes excessive "runout" an the wheel o.d. (wheel gear o.d. too).
You need to measure/inspect/compare the wheels, not assume. Also measure from surface to frame bottom for corners to sit level (or close, the machined axleholes are the real reference) Just as likely the track isn't level.
Gearside especially, the wheels must be true for good, even,gear mesh. A bent axle on the plain side might be straightened añd live many many years. I've simply used a vice before (& wood in the jaws ) À bent axle on the gearside means gears will go tight, then loose until it wears in (closer to worn out )
But the axle is more likey the issue than a wheel (assuming quartering is ok too; etc)
Gear/post being loose is just as bad as a loose axle bearing, the usual cause of gear screeching that usually notes an upcoming binding issues; a sandlike hesitation in the gearing, etc.
So, I'd look at my bushings, gear ,and post first. If the wheel gear teeth don't look worn over/soft edged, tracked, spotty wear, concave, etc. under close inspection, it's likely fine.
Ideally to check for wheel trueness you run a dial gauge on the wheel o.d. at 90° to the axle, then on the wheel face at the o.d. parallel to the axle. (and "in and out" axle slop is "thrust clearance") Doing it at 0° then 180° (or), 90° , 270° shows bore alignment if the frame can be referenced.
Feeler gauges or homemade go/no go blocks can also reveal issues with trueness just as well as a fancy dial