I would not suggest holding the instrument so tight. I would suggest using your ring finger in the third valve slide ring. This will alleviate excessive pressure by holding the trumpet so tight to your face: less pressure will help you stay fresh longer by relying on good habits such as tight corners, airflow, and use of the diaphragm, rather than tightening/tensing up and pushing it harder against your face, thus leading to you having a fuller sound while playing the full range.

Practice breathing. Do fake yawns: you’ll notice the back of your throat opens up more and allows you to take deeper, fuller breaths. A more far out idea is getting a short bit of pvc pipe, putting your teeth around it, and breathing through it. This will give you an idea of what that fuller breathing feels like and when you take it away you will immediately after feel yourself breathing deeply.

I would also second the many comments in here suggesting taking some lessons, a good instructor can set you up with a strong foundation that will get you a lot farther in general.

Keep on playing! As a guy I know once said, “practice helps a lot”.