I'm glad Andrew was able to offer some advice. I have no experience of drug testing I'm afraid so couldn't be much help but I have lots of experience of discretely managing peeing difficulties in the workplace and managed to do so for many years.
I do have a bit of a "turning the taps on" habit as you mentioned - it can be very helpful at times can't it but as you point out, it's not always possible so is not a very good way of coping. I find it useful to always carry a bottle of water with me to provide my own running water when needed - doesn't always work but it's helpful at times.
I do hope you're managing to sort out your employment - all the best.