The plastic stuff needs to come off. If it is riveted on (96 GSX should have had rivets mats like the green 96 GTX shown in these photos), then the rivets need to be drilled out. There will be rivet holes everywhere. My yellow HX had really thin mats, but they were stick-ons with no rivets - even easier!
Then epoxy the holes. Gravity will make this a pain in the neck. So you will have to do 2-3 applications with hardening in between. Silicone or RTV is not a good idea because the new mats' glue doesn't stick to that stuff.
Use degreaser or acetone to clean up the surfaces in preparation for the contact cement. Contact cement is all that in needed (both on mats and the hull) with some tacky set up time.
One tip : do not pull hard when applying the mats. This causes them to stretch. Then you get some overlap between sections you weren't expecting. So get a helper to make sure the panel doesn't get away from you.
The older Hydroturf (circa 1997) seems like it was a bit harder, so I think it wears a bit better than the newer squishier stuff (kind of like Pirelli P-zero high grip tires wear faster than boring high mileage tires).
The Hydroturf feels great even though it seems thin. And it makes a big difference on traction. I just about twisted my ankle on the first outing because I was so used to a some sliding in the footwells. But it looks and feels great.
Mats are an excellent off-season project. You can take your time, enjoy a few beers, no pressure and see immediate results. Oh, I think black mats on your GSX would make the boat look years younger because it has a more modern look.