'tushy' (or 'tush') is slang for one's buttocks. It's not considered a vulgarity - on the contrary, it's considered to be more along the lines of 'baby talk' or overly-polite commentary.
She is referring to the cleanliness (or lack thereof) of the public restroom, claiming that if the restroom is in such disarray that it lacks toilet paper/tissue to place on the seat before she sits down, then she won't sit down or use the facilities.
Hope that helps...