Door unlocked: the common source of anxiety is someone barging in, and you feeling embarrassed being there standing up. So firstly it is very common for men to use a cubicle for peeing, whether cos they want some privacy or it is too busy. Secondly anyone barging in would retreat immediately with a "Sorry".
Now you feel you would be questioned "why are you in there?" You wont, but have an answer prepared (which you will never have to use) like my answers to Dylan above. "Too busy out there" or "I prefer cubicles".
Secondly: to bridge the jump from locked to unlocked, use the faking technique. When you don't need a pee, go in the cubicle, leave door unlocked and do everything as normal except for peeing. This is to get used to being in there, without the pressure of having to pee. Repeat this exercise several times; because nothing untoward will happen, you will get more comfortable and will decide for yourself when you can introduce peeing.
I think the busyness puts them high up the hierarchy. Normally too much misfiring shows one is too high up the hierarchy; but you seem limited as to where you can desense. But see how faking helps first.
Responses are not allowed!