Yeah that trim will unsnap from the trim clips. I use a L shaped pry bar. You want to make sure you only pry right at the clip (locating the clip can be tough since it's covered by the trim and you can't see it... You just have to feel it out). Go easy, find the clip, and pry the trim out. Don't go crazy prying on the edge of the glass, it doesn't take much to bust a tempered back glass if you get right on the edge of it. Stick the short side of the tool under the trim and easily pry back. You might have to wiggle from side to side, but again... EASY.
As far as your problem, lots of guys install the windshield and rear glass using butyl tape. That's a bad idea. When that stuff dries out (and it will) it's going to leak. I'm not saying that's what you have there but if it is, reinstalling the glass using urethane sealant (like what is used on modern cars) will solve the leaking problem for sure for years to come.
I see WAY too many guys trying to stop leaks in the front and back of these old cars using anything from silicone to tub and tile caulk (LOL!). They just try to squirt it in the cracks. That's RARELY going to work and just going to make a mess. It's not hard to pop that trim off, cut the glass out, clean the pinch weld, clean the glass with new razorblades and lots of liquid lubrication. Then (recommended) call a glass company that does auto-glass to come by and run a nice bead of urethane and help you set the glass. It won't cost much. A lot less than a new trunk pan or damaged interior.
You CAN buy the urethane sealant and do it yourself but that stuff likes to be warm when used and auto glass guys will have tubes warming in the truck. They know how to lay the bead right and NOT make a mess while still getting the bead at the right height so that when the glass is set, your trim will be in the right place.
I did glass for over 20 years, just trying to help ya out and give you my $.02