Welcome, and congratulations on joining the
Scamp family!
You’ve nailed the two most likely culprits. An out-of-round tire or unbalanced wheel could make for a rough ride. When I bought a utility trailer years ago, one was so bad, that side of the trailer was actually leaving the pavement. It was new, so I had the whole wheel and tire swapped, and all was well.
An aged out torsion
axle, in which the internal rubber cords have become brittle, will also cause a very harsh ride. You can check that yourself by jacking up the frame on one side to see how much the wheel drops as the frame rises. Measure from the top of the wheel arch to the center of the hub before and after lifting. Let it sit a while and measure again. Worst case it could be completely frozen.
A rough ride can exacerbate frame stresses, and older Scamps, especially 16’ers, are prone to crack under the front of the cabin where the frame transitions to the tongue. Inspect that area carefully and repair/reinforce if needed. You’ll probably need to clean off surface rust (normal) to see the underlying metal.