First Aid will work well for you.
There is one in each park and they have rooms with cots of various kinds. Here is a picture of one of the First Aid rooms at the Studio.
Each First Aid has a waiting room with chairs where other members of your group can wait for you . You will need to check in and sign at the desk - they need to keep a log of people who used a First Aid.
It's not a big deal, they just need you to sign the log with your name and time. They need to know what sort of room you need so they can assign an appropriate one ( in some locations, some of the rooms are semi-private with 2 cots) and some rooms don't have a sink.
They will also need to let you know how to dispose of your used supplies and waste when you are done.
We have found the First Aid RN CMs to be be very nice. They will likely remember you if you are coming back several times a day.
They do rotate parks, so we were pleasantly surprised to be recognized by the nurse we had seen at another park at the Studio when we went there.
They can store some extra supplies for people, not sure about clothes. It may be easier to just carry those things with you.