Is Encryption without knowing the input directly possible at all?
What you're describing sounds a lot like mental poker.  That is, your Alice has a number of messages ("cards") that she wants to shuffle and deal to the other players, in such a way that she can prove to the other players that she does not know who got which cards.

Various protocols have been proposed for mental poker; see the linked Wikipedia article for some examples.  Apparently, there currently exists at least one practical and secure implementation of such a protocol.