My two blue crowns have a tendency to get a little chubby if I'm not careful. But unless something causes them to fly and circle several times, they usually don't get breathy. But I encourage them to fly both for the activity and help keep them slimmed & fit. A gram scale would be helpful for you to know if your's has too much weight. Somewhere around 180 grams give or take a few up or down.
For the beak, it depends upon how much your bird likes to chew on things. Tootsie seems to like the texture of things so she does not chew all that much but plays with them with her tongue. So every so often the vet will trim her beak down. Rosco likes to chew on every bit of wood or leather he can get his beak on so I never have to worry about his beak getting too long. So make sure your bird has plenty of toys to destroy.