I don't have a phobia of public washrooms but I don't like them lol. First I open the door if its a push one with my elbow or body or pull my sleeve down over my hand. The same procedure once I'm in the stall. If the toilets gross I hover lol over the seat - you need strong legs for that

or use toilet paper placed on the seat. I flush with my shoe.
At the sink I wash and if it's not an automatic water shut off....I leave it running until I dry my hands and use the napkin to shut off the water and to open the door or the elbow trick again.
I don't know if it helps knowing, we need a certain amount of germs to boost our immune system. It's also not good for your body to hold off to go to the bathroom. Now that I read my post, I kinda sound paranoid, don't I?