I don't know how old you are, but you need to get over your shyness, size doesn't matter. Maybe stop using the urinal and stick with a stall if you feel the need to "keep the thing hidden".
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I had to start using the shorter stand up because the water in the normal size one was just too cold [Coffee]
Just start using the ladies bathroom and the urinal issue becomes moot. :D
Personally I use a stall or keep the belt buckled if at a urinal.
Just start using the castle bathroom and the urinal issue becomes moat.
Reminds me of that Lewis and Floorwax commercial.
or this one
https://www.youtube.com/watch?v=R3YBLZMl03I
Never use a urinal in a public bathroom, carrying or not.
I can't think of a more vulnerable position to be in on any sort of a regular basis - pants open, tackle out, and strangers walking up behind you.
Stalls are a simple solution.
Change up your method. Couldn't find a non-bastardized version.
https://www.youtube.com/watch?v=SEJ-4cZ-cLQ