Sharon, the public restroom problem has been a problem for me too. Yesterday I went out to lunch with a friend and wanted to go to a shop afterwards, instead of driving back home immediately, so I decided I was going to have to really get something figured out.
I explained to my friend that I might need her help getting up off the toilet and didn't lock the door. Fortunately, although the toilet was a regular low one, the wall on my left had a great long grab bar. I sat down by keeping my torso straight, holding the bar and using my cane with the other hand sort of like handles on a chair. Kept my operated leg extended out in front so it wouldn't flex too much at the hip. Success!
Getting off the toilet was actually very easy too. Again, I kept the torso straight, kept my leg extended, and used the combo of the grab bar in one hand and the cane in the other to boost myself up without leaning forward.
So, I've found that I'll do fine as long as the bathroom I encounter has a good solid grab bar. The important thing is to keep your torso straight up and down and keep your operated leg extended.
Of course, that won't work everywhere you go, but maybe you can go on an inspection run to some of the places you want to go to check out if there are bathrooms in the area that are set up for your hip and frequent those places.
Your ability to walk normally without a cane is wonderful and amazing! I still use my cane for anything more than just a quick walk across the room because I find myself falling back into a limp. The leg muscles just aren't quite strong enough yet, but they are getting there
Dorothy