Struggling to replace a shifter

I'll preface this by saying that, while I've been biking for many years, I've not gotten into the gear/technical aspects of it before, even though I'm fairly handy around the house and with the car repairs. I decided to try to expand my horizons and make some changes on a bike and I'm having some difficulties.

Very recently I bought my first recumbent bike for $300 on Facebook Marketplace. It's an Actionbent, not sure of the model (see 4th image). The right hand shifter for the read derailleur (see 2nd image) seemed a bit wonky. I'd shift into the lowest gear and it would eventually just shift itself into second. I also didn't really care for the feel of the twisty shifter (see 1st image). I wanted to replace it with a lever type shifter (see 3rd image). Here's the link to what I got on Amazon: https://www.amazon.com/dp/B083F6SHK6

I watched a goodly number of videos online and still can't seemed to get this thing set up right. It just won't shift cleanly through all 8 gears. It seems I'll get it properly on 1st gear, it'll shift to 2nd, then to 3rd, and then the click to 4th does not change the gear. I click again and then it'll upshift. Eventually, when the shifter says 8, the derailleur will still only be on 6. Or, if I adjust it starting on 8, it'll downshift to only 2rd or 3rd. I can't seem to get the full range out of it. I've fiddled about with the L and H screws on the derailleur itself with no joy. At this point, I've futzed with the cable in the clamp on the derailleur so much the cable has frayed and a few strands are poking out. I've probably ruined this stupid (but thankfully cheap) part.

I just want to ride this bike, not spend hours in the garage trying to make this shifter work. Yet, now it's become an affront, a real challenge. Why can't I get this? This should be simple! I'm on the verge of taking this damn thing to my local bike shop but I do want to be mindful of expenses on this, my third bike. The wife has already been extremely open-minded.

Did I buy an incompatible shifter? Could that be what's happening here? Any consolation, advice, mockery, pointing out of the obvious you care to give would be most welcome.