Unless I'm missing something, I don't see any way to check which rooms are fully staffed other than going through them one by one, which is a pain. There should be some kind of a table where you can see which rooms are understaffed.

Is there a view where you can see which rooms are understaffed?

Upon double tapping on a room, you'll get a clipboard(i think) icon on your lower left side, clicking on this open a assign menu for the given room giving information on how many is working there, how many the max limit is and character names and specials.

Right, I've got that. What I'm looking for is a list like the dwellers list where I can see multiple rooms at a time, instead of going through them one by one.

No, that would make it to easy. One work around I do is just grab a character and drag them over the rooms. If it says+ or - it means it is full, if it just shows the number like A 8 that means it wasn't full

Ah, thanks, Superdeluxe -- that's the kind of tip I was looking for. I wasn't aware of the +/- thing. That's easier than clicking on a room and then clicking on the clipboard.