hey cool, you've got the right problems. LOL
well at least it shows you're using the right technique to sing with your breath.
but what you want is not more physical stamina, but vocal stamina. You must know how to maintain the right singing posture and technique throughout, that would depend how you sing in top condition.
A good idea would be to do lip thrills, I don't know if I got that word right. basically, you close your lips lightly, then you blow air through them, making a buzzing kinda sound. I think you get what i mean, just thrill and "sing" something without words. Using the "bluhbluhbluh" sound to sing a tune. Make sure your lips are relaxed and your air flow is constant. Keep the tune legato, smooth and flowing and don't stop. Try and get the air flow as streamline as possible.
Do that for a few minutes, running from high notes to low then up again, sing scales, all using the lips and lungs only. That should open up your airway.
You can also try pressing down on the gap between your two collar bones with your thumb, i'd write more about the pressing next time.
hope this helps.