The pockets theres no reason for those shirts to be stained from other shirts.
don’t know if your doing this but i have been told that some fabric softeners can have this spotting effect.
The spot with some bar soap think ivory works best but just anything basic really and rewash.
I guess you most likely use liquid laundry detergent. This could be the cause of your oil stains
For few weeks and solved it by changing liquid detergent to powder.
For few weeks and solved it by changing.