I can't see the splines well enough. And it's hard to tell cog wear by eye.

If it's noisy & skippy, then it sounds more like an adjustment problem, although it can also be worn. You can try a new chain. If you are lucky, then the freewheel is not worn too much and the new chain will work. But you need to have it adjusted properly so you can tell if skipping is due to worn freewheel and not to bad adjustment.