If your asthma is under good control, then getting winded WILL NOT bring on an attack, and if you have confidence in that, you will relax, and it will all seem so much easier.
First stop should be your doctor, or even a respiratory specialist, to ensure that your asthma is under excellent control. there are specific asthma medications that are most effective in exercise situations, none of them banned drugs in sports (at least in Australia).
A number of Australia's olympic swimmers have been Asthmatic, and it hasn't stopped them. this sounds as much a medical and psychological question as a physical training one. Fix the root cause of the problem, which is your anxiety that your asthma will trigger.
Best of luck.