Sorry to hear about your asthma, I have very mild asthma and the cold weather makes it worse, so I do get a bit scared if I am outside too long as it's harder to breathe.
I hate when my chest feels tight, I notice the difference between asthma or chest infection tightness versus just chest tightness from anxiety.
I don't really have any advice to offer, but do go back to the hospital if you need to.
Sorry you aren't getting as much support as you need, it can be very frustrating.