Sometimes my brain (maybe because I'm not a native English speaker) tends to come up with logical extensions to common formats; in this case, more than once I caught myself thinking (not out loud, thankfully) that I was feeling "pissy". Now, the actual meaning of that word is nothing close to what I meant, but I can't help but feel like there's a gap there. Other people seem to have wondered the same but I don't think I can draw anything conclusive from that...
A quick Google search has not revealed to me whether or not this example is only used in Australian English and whether or not it is slang (because it is certainly not formal and you wouldn't say it to your teacher or employer).
Sometimes (at least in Australia - perhaps those from other Anglophone countries can tell us whether they use it too) we say, 'I am busting to go to the toilet'. 'Busting' means that you REALLY need to urinate.
The question asks for a word like 'hungry'. Well, this word is more like 'starving'. So it does imply quite an urgent need to urinate.
If you wanted to be more polite I suppose you would just say something like, 'I need to go to the bathroom'.
Yes, in medical terms we call it "urinary urgency" - this implies the need to urinate and implies the frequent need for urination. However it's a medical word - you can say that the patient suffers from urinary urgency. You can't use it to say "I feel like I have urinary urgency" - doesn't sound right.
To micturate is to urinate; dictionaries say micturate derives from the Latin micturīre, "to want to urinate."
So you can combine the old Latin root, and the -y noun suffix (which means "full of," e.g. dirty means "full of dirt"), which gives you mictury. ("Dude, I am seriously mictury...")