I have this exact issue! Plus I am somewhat claustrophobic.
I was able to work it out during my pool dives with a lot of time spent just concentrating on buoyancy... hovering close to the bottom without touching and using my breathing to hold or change my height in the water. When I was concentrating on all that, I was able to not be as concerned about the breathing... but if I started thinking about it, I would begin to panic a bit... So I ended up breathing a lot during all this. I emptied a tank of air to 750psi when everyone else had 1500-1800 left on day one. Then on day two I emptied it to 500psi and we still had another 30-45 minutes of work to do in the water and had to swap tanks.
At that point, I started really concentrating on my breathing, attempting to slow things down... and the more I just made laps around the pool slowly, working on breathing slow, the easier it got. That last tank I ended up using much less air than previously, and I had to do the share air as donor and receiver, controlled emergency ascent, and retrieve unconscious diver all during that last tank.
I had other issues at my open water checkout but I was more relaxed about the breathing on the one dive I was able to make once I had a few minutes to concentrate on breathing again.
I sure hope this gets easier to deal with over time and more experience.