But I can understand your want to swap the engine, depending on how far you want to go, you could source a 2.0 bottom end, get that refreshed and then spend a couple of weekends swapping your head over and putting it in.
Though I'd still get a leak down done on your engine first, no point in putting a potentially worn head on a shiny new block.
Problem is nowadays, any engine you swap in is going to be over 30 and feel almost just as tired.
Personally, I'd use the money getting your 1.8 rebuilt. As long as the bores are good, a hone, fresh set of piston rings and a handful of other refresh bits should see it back to it's best.
First, you should get someone to do a leak down test. You may find you've got something like worn stem seals/guides and can get away with just a head rebuild.
I know I'm probably the last person to tell you not to swap out the engine, but if you want to get best bang for your buck, you may be better off keeping what you have.
Do you have a multimeter? If so, take the boss off and measure continuity (resistance) between the ring at the back of the boss and its chassis. If you have 0 or very low ohms, your issue lies in the boss and not the method you're using it to fix to the car. It's entirely possible something has moved in the boss and is constantly earthing out the horn push wire.