Spots on cotton shirts after washing them that were not there before?

I have a chronic problem with my laundry. When I wash cotton t-shirts they often times come out with spots on them, the spots look like they are a wet spot when they are dried. I have a new washing machine so I don't believe it's that. I use high quality soaps and sometimes spray the shirts with spot removers even though they have no stains just to avoid my shirts coming out with these spots on them after washing. I've tried it several ways with no luck! Can anyone chime in with a solution before I ruin anymore shirts?