Bought a house built in the 70s by the original owner who I bought it
off. The bathtub and sink were both dripping, so we went in and replaced
all the washers.

I messed up (or the threads were stripped) since when I put in the inner
parts, it just kept screwing as if it jumped off track. So if I screw it
all the way in, it shreds the end washer again.

So I only partially put it in, its not fully tight, but it seals shut
(the tap doesn't drip). Now the problem is when I turn the water on,
some water comes out the handle. Not a great solution at all.

So, other than taking all the shower tiles out, and cutting the old
copper fittings and putting a new one in, does anyone have any ideas on
how to fix this?

I am going to try some teflon tape to see if it helps seal it better.