Can people with depression get moody, irritable, and upset at even the most minor things?

In a word, yes. Pressures on the job can also add to the anxiety. Maybe you need a hobby, or take up knitting or get a pet or something to provide you a means of escape during times when you need it. It might work. Crying to relieve tension is not a bad thing, but at the same time face issues with an open mind. Take the bull by the horns, face up to the challenge. Things almost never get as bad as our fears make them out to be.