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.
About the shower, I hang my towel just outside of the shower and get it before I step out. I get myself mostly dry before stepping on to the shower mat.