I found that chewing gum with my mouth open helped me breath better. Not so glamorous, but I was less inclined to hold my breath or gulp air while chewing my cud. These days I have the opposite problem. Instead of being too tense, I'm too lazy. Apparently it's not cool to yawn while playing drums.
If you sing or count our loud (the louder the better) I think it will help you breathe properly. I remember seeing it in a video - a drummer was talking about how drums doesn't force you to breathe when compared with other instruments such as woodwind/brass instruments. This will also help with your timing. Also check your posture as well because if you are slouched over you will be putting pressure on your stomach which might be causing some of the issues as well.
I found breathing with nose only and reducing unnecessary torso movement help a lot. But singing might not help, as I have that problem when singing too. Maybe I should breath nose-only when singing too?
Just do what works for you. If you become more conscious of your breathing it should help you relax while you are playing which is the goal anyway. Good luck. I will also try to start thinking about my breathing while I am playing and see how difficult it is.