OK, if you've got a 126 wheel in there, yeah, 7 speed should fit. Perhaps a washer needed as above, but more likely not.

Try the Sunrace. Modern shifting and durable. I've ridden quite a few miles on one. I'd only put one on my Peugeot to get rolling for cheap while my custom bike was getting built. Surprisingly good IME.

You'll need a new chain at the same time.

Wear might be the issue with your shifting, but there could be other problems. Typical symptom of a worn freewheel is that it start to skip teeth in one gear. Usually happens while climbing hard or otherwise feeding it a lot of torque.