A: To refresh: the Action Gift Giveaway (aka Saturnalia in Olympus Fluxx, or All Must Have Prizes in Wonderland Fluxx) says (with some paraphrasing, since apparently there are two slightly different wordings):
“All players must pick up one of the Keepers they have on the table (if any) and hold it aloft…. [The Keeper] will then be placed in front of another player… Players without any Keepers on the table must be given gifts first.”

This is actually tricky, and we probably could have been slightly clearer. If you are still holding a Keeper in your hand, waiting to give it away, you are still considered to be in possession of that Keeper. This does mean, however, that once you have given the Keeper you are holding away, if you don’t have any other Keepers on the table, you are now someone who must be gifted first.