Compatible thumb shifter for Shimano Nexus 8 speed internal gear

You could cannibalize a barrel adjuster (I think that's the broken bit if I understand your description) from an old shifter or derailleur. For most of the time these adjusters have identical threading. Try your LBS, they may have a parts bin with bits and pieces.

Answer: NO

I did the opposite once - bought an 8 speed Nexus shifter that was super-cheap, for use on an 8 speed derailler. The cable pull of a nexus/alfine shifter is different to that of a conventional shifter, so clicks will not line up with gears.

You might think that a friction shifter would let you get it in the right place, but since the IGH is internal you won't hear the gears meshing/clattering if wrong, so there's no way to know if you're in-gear properly or just riding on the corners of teeth. This is very bad for the gears.

Your two options are to replace the shifter with a working one, or repair your existing shifter as per @carel's advice (which would be my first option)

As for sourcing parts, I've had good luck buying "Shimano" parts from aliexpress. I bought a chain tensioner from the LBS, but they renegged saying it was no longer available. So I found it for 1/4 the quoted cost, and its identical to the original. Time to ship can be high, so maybe order one and while waiting, try to repair your existing shifter.