Unfortunately, using such a system, Alice might worry that Bob could show others her messages and signatures, thus proving to others that she wrote and signed them. Such worries are not unusual if someone goes to the trouble of encrypting and signing their messages.

Incidentally, this has actually been a practical concern on the black-markets: there have been occasional attempts to break pseudonyms by sending them messages encrypted to other pseudonyms' keys, and the initial proof that Dread Pirate Robert 2 was StExo come from DPR2 accidentally releasing a message signed by StExo's key.