I was not happy when I saw all the ruined clothes. Surely there was no way to fix those stains.
Then, along came my good friend GOOGLE! :)
Turns out liquid dish soap is the BEST stain remover on the planet. I pre-treated each and every pinky-red stain with a generous amount of dish soap, washed the load of laundry in the washing machine again....and voila! Every single stain was gone.
It still isn't a good idea to wash and dry your tubes of chapstick. But at least now I know that it can be fixed!