If I were you, I would ask your instructor to help you conserve your energy-- where should you be relaxing, where should you maintain some tension? I think you'll probably hear something like-- don't tense in your legs/seat, but do focus on pulling your shoulders back/sticking your chest out to keep your posture straight. I know how much tensing in your seat and stressing out about being "floppy" can wear you out!
I think it's a great idea to take lessons on a different horse. :) That way you can set a precedent with your own horses-- when you do ride, you'll ride more correctly and they'll learn to respond to the correct cues and body language!
Maybe you can start going for walks every day or doing squats/situps before the shower... little things to help strengthen the muscles that will keep you in the saddle so that you become fatigued less quickly. The other thing-- after a lesson, go home and eat something high in protein! Eggs, meat, etc... the same way you would as if you had just worked out (because you did!).