I have a Dual Exposed Thermostatic Shower valve. It does not leak. The temperature is fine. The on/off lever handle however used to hit a stop point at off, and now does not - it goes around forever in circles (turning the water flow on and off as it goes around). It's fine for now (working OK, I drew a marker pen line to indicate when to stop turning to shut the water off), but it will need fixing. Is this a spare part I can get, or do I need to replace the entire valve?

Replace temperature handle.
https://www.youtube.com/watch?v=r2lcERldHe4

You are not saying what brand / model number faucet it is. There is a "stop" under the handle. In some cases, the stop just got lose and needs to be pushed back into the faucet. In most cases, the stop is broken and cartridge will have to be replaced.

Turn off water for the whole house, remove handle and see what type of stopping mechanism you have. Proceed with repair / replacement accordingly

Can you post a clear photo of the faucet in question ?

Milo