I got a puppy and since then have been short of breath. Am I allergic to her?

It's Possible. Ask your doctor to test you for allergies. A simple trial is to bathe the dog twice weekly to keep the allergen down, keep the dog off your bed and out of your bedroom. Consider kenneling the dog for a few days or giving it to a friend for a week and see if you do better.
Either that.... ..Or she's so cute she takes your breath away. See your doctor to find out which.