You don't really have to rinse it out every time. Even if there aren't any people in there, I'd be afraid of someone walking in on my lol. Urine is sterile and as long as you clean it at the end of the day you're probably fine. The main thing is to make sure that you drain the stp as well as you can so there isn't a lot of leftover liquid there. One or two drops isn't going to do much. I use a funnel type and personally, I kinda got used to a slightly damp feeling pretty quickly and it does dry.
There's a lot of different kinds of stps out there, it depends a lot on your budget and what you're looking for, like a packer/stp or just a home-made kind, funnel, medicine spoon etc.