They don’t sell hygiene products in ladies’ rooms anymore? Just check out your average big employer. I’ve never worked in an office that didn’t sell them.
Restrooms that I have seen do not have vending machines, and I would not be embarrassed to use one in front of anyone if they did.
Do you know any transgendered people? You need to expand your horizons. Most transgendered people that I have encountered are very meek, self-conscious and retiring. They aren’t going to hurt you in any way, except maybe the masculine lesbians, who use ladies’ rooms.
If you are so frightened, maybe you are better off avoiding public restrooms.