I'm a shaker. I stir too, but I've solved the sticking problem by taking the cardboard disc, once it's stuck to the rim, prying it off and sticking it back into the lid upside down. I think I started doing this when I was 10 years old (circa. 1973) and haven't had a problem since.
Funny thing, I've still got some 25¢ Testors brand model paint from the '70s that's still good as gold. Yep, cardboard been flipped.