Shifter only uses first 4 gears
If you pull the inner wire by hand while spinning the pedals, does the rear derailleur move the chain to all 8 cogs on the rear cassette? (suspend the bike in a workstand or hang from something with a rope to make this easier)

Does the shifter (the part on the handlebars) click through 8 different positions as you press the buttons?  Can you feel it pulling the inner-wire a little on every click?

Sounds like the shifter unit is pogged up and some of the pawls inside are not latching into the teeth.  You could try disassembling it, clean all the old lube out with a degreaser/petrol and then re-lube and reassemble.  I've done this with brifters and it was hard work but fixed the problem.

If that doesn't work, you're probably up for a replacement 8 speed shifter... not too expensive new.  Your local bike coop might be able to help too.

Fit a new inner cable while you have the chance - they're cheap.