It's normal for her to breathe heavy due to stress, but it should only be when you are near her. If she is trying to sleep, and having breathing problems, something might be wrong.She should be exploring her cage though. My lovie explored every nook and cranny of his cage before settling on his favorite perch.
I hate to worry you, but I'm only speaking on my personal experiences. My little budgie never had any mucus or abnormal poop. He was active and playing with toys despite being lethargic and struggling to breathe, and he ate right up til the hour he passed.
Perhaps your best bet is to keep a very close eye on your new budgie, and monitor her behavior. You should see to getting a vet visit scheduled ASAP. Sick or not, it's always best to have one scheduled so that you can have reassurance from a certified avian veterinarian.
I think you're just stressing out over nothing, but it's best to be sure.