Can exercise-train while I am trying to overcome skin fungus?

Yes. Skin fungus should not inhibit your ability to exercise/train. Just make sure, though, that you are sharing towels, etc. with other individuals as to not spread the infection. If you have active skin lesions, then keep them covered when using shared equipment.
YES. ALONE! You should not utilize common areas in a gym such as common showers etc etc

Hope this helps
Dr Z.