I’ll go first. After your turn the water off in the shower but before you get out, use your hands to wipe off any standing water on your body. Maybe even give your legs a bit of a shake. This way, you won’t drip nearly as much when you get out, keeping the floor and your towel drier.
Surgical cloths. They have to be disposed of if they were even in the operating room. They still sanitize them though. You can probably find them for free, but if you pay any money it will be incredibly cheap. They are very low-lint and have a coarse texture that makes them perfect for cleaning.