You will find in the US that it common to wear a vest/T shirt under a normal shirt especially when its hot. The undershirt takes the sweat off the body and stops the outershirt getting wet.
I always wondered about this but remembered that when my dad was stationed in Malaysia, he wore a vest under his uniform green shirt (he was seconded to the RMAF) and never had a sweat stain.
Dont worry about the Transair shirts but nip to a department store and buy some good undershirts.
My last lot I bought in a mall in Florida. Forget thick heavyweight and expensive white shirts solutions to what is a similar problem.