Are your school custodians friendly? If they trust you not to abuse it, a $20 hand shower attachment turns a mop closet with utility faucet and floor sink into a shower stall.
This is a good option. Rubbing alcohol applied at full strength might cause more issues than its worth. Anything from dry skin to the most extreme alcohol poisoning. Is there a middle school or high school close? When deployed in the military, baby wipes are all we had sometimes, it helped but was by no means a cure all for stink. This may sound gross but personal trimming of hair, under arms and groin can help with odor control. I'm not saying you have to go clean shaven but hair does trap odor. Other things that can help reduce body odor in the long run, cut out smelly foods, shower night and morning, incorporate more zinc, magnesium, and chlorophyll into your diet. Hope that helps, seems like you have a ton of info to go through. I would think in Beaumont Texas you would sweat your tail off most of the time anyways.