You can bring a mini TP roll made for hikers, a small package of tissues, napkins from fast food restaurant, a folded up stash from a big TP roll at home, or use a bandana as a pee rag.
Pee rags (aka bandanas) are great for keeping it zero waste and having the luxury of wiping. Attach it to the outside of your pack or stash it away in a specific pocket of your pack.
Be sure to PACK OUT everything you pack in, including your used TP. A small plastic baggie works just fine.
Changing your tampon? Yes, pack that out too! Realistically, it’s more likely that an animal will dig it up before it decomposes.
Under those circumstances, save yourself from carrying around used tampons + plastic applicators and switch to the healthier, more eco-friendly lifestyle choice.
Additionally, another option here is the old school air dry method. You can do a little bounce up and down in an attempt to shake off any remaining drops.