Let's take a brief look at what changes when you play for an extended period of time: the pressure of the mouthpiece and the tension of your embouchure muscles limits the flow of blood, without blood, the muscles have no fuel to do their job and they get weaker, causing the player to use more pressure - or get an airy tone. You have the solution, you just don't realize it yet. When your teacher stops you, and then you continue after a short break, it works again. You need to find ways to remove pressure/tension more often. With practice, 2 beats may enough to regenerate. there are usually plenty of musical opportunities to relax. You need to use them. Take the mouthpiece off of your face when you are not playing. My recipe: do the right things for endurance like lipslurs and long tones as well as increasing your musical view. Teach yourself to look for opportunities to "rest". With practice, the break time required gets shorter and shorter as your endurance increases. Smart playing is the solution.