The mountain bike might be the better choice to get started on, I suspect if you try bike touring and like it, you will end up buying a new bike later. Starting with what you have makes sense. If the Schwin is mid to late 90s, it would likely have an 8 speed cassette that would work well for bike touring. Even if it is 7 speed in back, that is still quite usable. If it was a true mountain bike of that vintage, it should have a triple crank, thus has wide gearing which is good. And you have had it long enough to be familiar with it.

The chain stays are the metal frame tubes that go between the rear wheel and the bottom bracket. The bottom bracket is the spindle and bearings that your crank rotates on. Longer chain stays are better for touring, and if they are quite short then your panniers have to be mounted further back so your heels do not hit them, and putting the panniers that far back with the center of gravity that far back can impair handling.

If you start out using backpacking gear, that is light so you are at an advantage of keeping the bike light. I also backpack, but I am a lot less weigh conscious when the weight is on wheels instead of on my shoulder straps and hip belt.

Tell us more about the mountain bike, gearing details, tire sizes, model, etc. Seatposts can be shimmed if you need a different one.