I went to the derm for my excessive sweating and one of their recommendations actually ended up helping the stain issue. Bathing at night, make sure underarms are super dry, then apply the antiperspirant. It needs to sit on dry skin for a bit for it to work its best. By the time I get dressed, the antiperspirant has soaked into the skin and I notice I have less of an issue with it getting into the arm pits of my clothes.

TLDR: put your antiperspirant on as long before you put your clothes on as possible. Using a fan or hair dryer can likely speed up that process when in a rush.

And another: buy only flat sheets, one size bigger than your bed. You won't have to worry about having the right sheet, and one sheet doesn't get worn faster than the other.