Did he get another staff? If so, did he make his own?
Did he have extra staves just lying around in case this exact situation occurred?
Did he become staff-less?
This scene doesn't appear in the books, and Radagast never appears in the films after this point. As far as I know, there are no interviews that reveal what happened.
This is just one of those elements introduced by The Hobbit films, and never followed up on.
Any of the explanations given in the question are plausible, but there's no indication that any of them are more true than any others.
It's possible this is the films' way of incorporating the idea that Radagast turned away from his mission:
Indeed, of all the Istari, one only remained faithful, and he was the last-comer. For Radagast, the fourth, became enamoured of the many beasts and birds that dwelt in Middle-earth, and forsook Elves and Men, and spent his days among the wild creatures.
Unfinished Tales Part IV Chapter II: "The Istari"
In which case he wouldn't have needed a staff; but this is only a theory.