Hi, I have some experience striking, but am mostly a grappler/judo man so, with that in mind, I have a couple things to add. First, I believe that most people are more relaxed during conditioning, and more anxious during actual sparring/rolling and competing. To get an athlete to relax during the actual bout is definitely a coach's true challenge, you are not alone in this regard. That being said, breathing is one of the main keys to becoming more relaxed, and being relaxed is important for maximizing efficiency of movement. Breathing is an effective tool for stress management in life as well as sports. You can usually hear good coaches in the corner of mma fighters, grapplers and strikers giving specific breathing coaching points during the bout and in between rounds. I recommend a higher volume of sparring/rolling sessions, and mit sessions that are conditioning based, with a focus on efficiency of movement combined with effective breathing.
Regarding breathing while absorbing strikes, especially to the body,: Yes, exhale. You may be exhaling a little too much volume if it is inhibiting your breathing during the round in general. So, monitor your volume. Also, keep in mind that there are times during the bout to take deeper breaths. Be aware of when those times are, and make efficient use of them.
Hope that helps.