Shamir's Secret Sharing - Impersonating the Dealer
Yeah, as stated, that's an unsolvable problem.

Basically, if you assume that the dealer loses everything that would let the shareholders (or anyone else) distinguish them from some random stranger, then there's nothing they can do that some random stranger couldn't also do.

The typical real world solution is to let the shareholders identify you by some physical means, such as by your fingerprints, iris patterns, DNA or just general familiarity with your physical appearance and personality.  Such biometric IDs are hard to accidentally lose (especially if there's some redundancy) and even harder for someone else to copy (at least in a way that would fool a careful human inspector; unfortunately automatic checks are often quite a bit easier to trick).