My bike has an 8-speed nexus hub and the plastic adjustment screw broke. Now I can't adjust the cable anymore and can't use some of the gears. My question is, can I use a normal thumb shifter on my nexus 8 speed since the nexus shifter (internal gear bike in general) is very uncommon where I live and I don't want to pay for the ridiculous price of shipping fee plus tax on my old bike.
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.