FWIW, Birkenstock used to (& maybe still does) sell stuff to brush on cork soles probably to keep it from crumbling. It seemed like it was identical to craft glue -- like Elmer's. It dried to a bit of a low sheen like white glue. Maybe you can try that as a stop-gap measure?

I was expecting something that looked thrown together, but this is waaay too fab for that! terrific outfit. Sorry about the shoes and lack of free parking.