Are you able to get more than one of each shirt? I like the gusset idea, but if for some reason that doesn't work, maybe you could cut up an extra shirt to remake the sleeves.
Getting your own shirts embroidered might be an idea, although it might get expensive if you have to pay design and setup charges. If you can use the same embroidery company that your organization did, and just provide the shirts, maybe they'll charge you a nominal fee.