I'm another asthmatic. What you're describing sounds a little more to me like the type of not-being-able-to-breathe you get during a panic attack. (Or else, the baby is really crowding into your lungs, but that doesn't seem like it to me.) It doesn't seem possible (In my not-at-all-professional opinion) that you're having actual asthma attacks *just* during contractions.
However, panic attack not-being-able-to-breathe and baby-crowding-your-lungs not being able to breathe are similar to asthma in some respects. Let me give you some tips that help me (as an asthmatic and as someone who has had at least one major panic attack):
Asthma is a condition where your lungs actually cannot expel the oxygen that's in them (because of muscle tightness and congestion). Anxiety can reproduce this muscle tightness in the upper chest (but you wouldn't get the congestion part) and make it difficult to breathe. Recieving oxygen would *not* be helpful in this case, because the problem isn't that you have no oxygen going in, it's that you don't have enough CO2 coming out! (This is why they don't give asthmatics oxygen, they put us on nebulizers during serious attacks, to give us medicine to clear the lungs of congestion, and relax the chest and diaphragm.)
So, even tho it *feels* somewhat counter-intuitive, focus on breathing out. Really use your muscles (diaphragm and chest) to expell your breath as fully as you can. The inhalation will follow. Don't try to force the inhalation.
There are lots of yoga breathing exercises that can be helpful, but mostly, just focusing on the out-breath is what helps the most, I think. (And it is super easy to do.)
The other thing you want to do (and this also applies to baby-crowding-your-lungs issues) is open up the chest and diaphragm. Roll your shoulders back. Try massaging your chest in an outward fashion (stroking towards your breasts from the center). Think "open." Having someone rub your upper back is REALLY helpful. (I personally find light massage to be the best, but some people prefer deeper massage). Water is GREAT! A hot bath will relax chest muscles really well.
Asthma attacks usually leave me with an extreme back ache from all that concentrating on breathing. That's where the massage and hot bath really come in handy. Also, muscle rub (tho I'm not entirely sure I'd want to use that during labor...).
There are also herbal remedies for relaxing the chest muscles, but I unfortunately don't know too much about them (since I use an inhaler). I do know caffeine is a good "OMG, I lost my inhaler, what do I do?" works in a pinch substitute. I'm not sure if it helps with the congestion or the muscle spasms, tho. Plus, drinking a strong cup of coffee during labor sounds really gross to me... but maybe a little tea? I don't know, it's just a thought... (then again, caffeine agravated anxiety. Maybe herbs that help with anxiety like Valerian and St John's Wort would be more useful. Like I said, I don't know much about this subject...)
BTW, heart palpitations are also an indictor for anxiety. Is there someone you can talk to (a therapist or clergy person or very trusted wise friend) for help with this anxiety? Did you know anxiety is the #1 psychological disorder in the US? It is very common, there's absolutely no shame in having anxiety. Getting help is a good thing
Even without meds (or with meds, if the anxiety is severe enough) if you can find someone to talk with, you can usually see a huge improvement!