Its a big jump to using cubicles with the door open for peeing. The result is as you say nerve-wracking. There is an intermediate level, which is to ensure you do not need to pee, and to go in and follow the peeing procedure, but not peeing. You may say this is odd, but the reason is to reduce anxiety to a manageable level. If you can do this "faking" until you become more comfortable in a cubicle and not peeing, you will then feel more able to start drinking and peeing in that situation.
The theory is that the anxiety is due to two things: one the need to empty one's bladder, and two the situation: i.e. everything happening in the toilet. By removing the anxiety of needing to empty the bladder, you are left with only one source of anxiety to manage.
The beauty is that you can go back in every say five minutes and repeat the exercise, and you may well find after the fourth time that your anxiety has significantly deceased. However, just repeat as many times as needed. It is a concentrated, effective and proven desensitisation.