Puffy welts is likely hives - which is an allergic reaction that has turned severe. You need to keep him dosed on Benadryl until you can get to the vet - at which point, the vet may want to give a presrcription antihistamine that is stronger and/or a steroid shot.
Did the rash start in a certain area? An area that would be in contact with the ground? He may have been in contact w/ a pesticide, a bug bite, or any number of things in the new environment. Keep him hydrated - but definitely keep the benadryl in him as much as possible to keep the histamine reaction at a minimum. Keep us posted.
