This might be the case as when you login with your alt account, the cookie of the previous account must have been stored on your machine and as you login with new account, the my bets tab might request for the cookie previously stored in your machine and then it displays the bet related to that cookie info.
Actually, primedice stores your login token in "access_token" stored in localStorage, not HTTP cookies. I'm honestly not sure why, I've asked here before but never got a reply. Anyway, they may indeed store recent bets in an HTTP cookie, but the browser login token is stored in localStorage in javascript.