Any thoughts on air purifiers and improving air quality?

@cneville1 You are facing a tough problem here. I don't know anything about your particular air purifiers, but I know a bit about how school districts handle contracts.

They typically want a bid for the least expensive one-size-fits-all solution, without regard to what is appropriate in each room. Then filters are installed without regard for proper ventilation/fresh air exchange or moisture management. The needs in a wrestling room are vastly different than a classroom or office. Staff receives minimal training, and there is no provision for ongoing maintenance, sometimes not even for a budget and schedule for replacement filters.
All of these factors are a recipe for exactly what you describe. My daughter, a high school nurse, describes wrestling rooms, weight rooms and locker rooms as germ pits, so close body contact and confined spaces contribute to the problem.
You need a thorough inspection by an environmental health specialist. District admin is unlikely to listen to you alone, but if you are accompanied by unhappy parents you might get action.
Have you approached administration with this concern?

We have approached EVERYONE with this concern. short of actually bringing in a doctor to explain the situation. All we know is that when the air purifiers are not running, we seem to have less illness. I also know that the filters do not get changed regularly unless by the wrestling staff and even if changed every other day, there is a mold growing from the humid air being pushed through them. I have pictures that I have taken and shown the admin but to no avail. Im thinking our next step is to consult a physician.

@cneville1 Humidity is an issue causing increased mold growth. Have they considered room dehumidifiers in addition to air filtration? if you have mold inside the air filter units, all of that needs to be cleaned and disinfected and new filters replaced otherwise it just distributes mold spores or dust, and of course filters must be replaced on a regular basis. You could take your concerns to a county health department, and they may render a decision on what should be done. That would kind of put you in an uncomfortable position. There may also be mold in the building itself and in their duct work and it may be a bigger problem.

I think consulting a physician is a good idea, however, their opinion may not matter to those in charge. I don't know the specs on your air filters, and there is varying quality of filtration. Some can filter out very small particles as well as pollen and spores.

One suggestion I had heard about is to construct your own inexpensive air filter with the best furnace filters you can buy along with a simple box fan called a Corsi -Rosenthal box. You could do that as a test to see if it improves the situation. You can learn about that in this post. There is a link to a website for the Corsi Rosenthal Foundation with free information on how to construct it. If you make your own box and are still getting mold growth with all new filters, then humidity must be lowered as well and/or it may be a building problem. Testing your own do it yourself filters would give you some control of what's happening and as well as being a test of the environmental conditions in your wrestling room. The discussion is about ultra purification with carbon and HEPA filtration and that may be interesting as well. The suggestion about the Corsi Rosenthal box was made by a member as an economical way to solve a problem rather than jumping into expensive air filtration units.
https://connect.mayoclinic.org/comment/1181180/