Breath control, breath control, breath control! I recall one of the top University Marching Band directors talking to us (Midwestern Music & Art Camp circa 1970 :-) ) about his "aha" moment. He was a trumpeter & got a summer job (high school or college age for him) with a circus orchestra. First couple performances, his embouchure collapsed half-way thru and he staggered to the endline w/ sore lips. Then one night he played the whole show without any lip strain or pain! But, in his words, when he tried to stand up, "I felt like I'd been kicked in the gut by a horse." That's when he learned breath control.
As to the noise problem-- nothing you can do other than mutes (yes, there are sax mutes!), a practice cube, or playing under a couple blankets arranged like a tent
EDIT: to expand on breath control as it applies here.
I once had a teacher (clarinet) who would poke me in the navel with a pencil while I was playing. In theory, if I had my diaphram muscles properly set, there'd be no change in my sound. Took a while :-) . In addition to this, learn to breathe "deep," inhaling from the lowest part of the abdomen while trying not to allow your shoulders to move. It's pretty much the opposite of what one typically does when "gasping for breath" during extreme athletics. That, along with learning to 'focus' the airstream (admittely a psychological concept that's not really related to the airflow at the mouthpiece), will let you maintain a continuous pressure head and limit excess airflow. That's about the best I can do in print :-( .