Even during very busy times, I've never found a stall out of TP. Funny, this is one thing that I've never thought about (and I used to be an excessive planner!).
I would just have a little pack of kleenex with you, as you said you're expecting. You never know when you need kleenex anyway. I also bring a few wipes, in case someone needs them for #2 or to feel fresh, sorry TMI!
Also, hands can get sticky eating stuff like french fries, ice cream, or those huge turkey legs my son likes! Definitely some fresh wipes are handy.