I am very familiar with breathing issues, both real and psychosomatic (which can be no less serious). I had asthma and upper respiratory problems for a long time until I was put on Advair (a steroid inhaler) for four years during which time my breathing and health was better but I had side effects and long term negative impacts on my health. When I decided to go off it for good my kidneys started producing too much mucous and I had chronic bronchitis and pneumonia for an entire year. Part of the solution for me was the drastically alter my diet in order to make things as easy as possible for my system to process without generating mucous until my body could recover from the effects of the steroids. But also essential was a more determined, resolute, and fearless attitude to being healthy and life in general.
The biggest key for breathing problems is to maintain natural breathing patterns. When a person suffers from a breathing problem regularly, they are likely to worry about it, and be on the look out for it. Then when anything does occur they immediately focus their mind on it, disrupting natural breathing. In this situation every breath becomes willfull and controlled by the conscious mind. This type of unnatural breathing causes a shortness of breath and increased tension. This in turn can cause even more anxiety and create a cyclical pattern. The person needs to just let go and breath naturally but this is not as easy as flipping a light switch.
As you said, the long-term key is to get to a point where you are no longer worried about your breathing, so that your mind will not be focusing on it, and your breathing can remain subconscious and natural even in times that would normally cause worry and anxiety.
Remaining busy and engaged will help in this. When the mind is preoccupied with other things then it cannot be so focused on worrying about breathing. Often times people say they are too busy to get sick, and there is alot of truth to that. Idle time is the worst enemy of someone with alot of anxiety. Activities do not have to be physically demanding, as long as they are engaging and constructive. Things that involve responsibility are best, such as business, work, volunteering, cooking, housework, taking care of children or family, etc. When you are worrying about the needs of others you have less time to worry about yourself. Owning a dog is also good, because it gets you outside for walks in the fresh air and sunlight and they keep you light hearted.
Making some sort of change to your diet, habits, or schedule is also good. For me when I changed my diet it was something to feel reassured about, in the way that the mind says ?I did this, so I’ll be fine?. Something like quitting smoking, eating less red meat, going to bed earlier and on a more regular schedule, cutting out sugar, not snacking, etc. All of those types of things will not only help with overall health and reduce stress on the body, but they are also something to fall back on and take reassurance from. It helps you feel like you have earned your improved health or breathing, which in turn increases confidence and decreases anxiety.
Yoga practice is also very beneficial. As for choosing a style, they are just based on the way a particular person sequenced and grouped the postures they emphasized and what guidelines they recommend. Hatha Yoga technically means any physical Yoga. I would recommend finding someone with obviously radiant good health and a happy demeanor. Clearly what they are doing works, and being around them will allow their care-free attitude to rub off. If going to classes wears you out, then try something where you are going at a slower pace, stopping to re-group between postures. Perhaps a DVD or something where you can stop and start as you please with no pressure. Maybe someone on here can recommend helpful DVDs. Don’t try alot of pranayam until you have first built up some lung strength through asanas first, otherwise it will just wear you out.
