All the above advice is excellent, I would also caution you that placing your eye s close to the rear sight can give you problems, propper eye relief on a service style AR15 is 1.5-4 inches. As a 6th award Marine Expert rifleman, it also sounds like since your sling is loosening, that you are tensing up your body to try and correct for the lack of sling support, this will push your strings low as your shoulder tenses, and right as you allow the barrel to go right with the lack of sling tension.
Without seeing your shoting position, its hard to give the best advice, but i also coach my younger Marines to roll their supporting hand further underneath the handguard, ensuring the handguard is laying flat along the meaty portion of the palm. this often will help keep the rifle from moving around during rapid fire strings. Good Luck!