Q: In Fantasy Fluxx, can one move the Dragon any time one one’s turn, or does one have to play another card, like Exchange Keepers?

A: On your turn, you can just move the Dragon to someone else, and then trash a Keeper or a Creeper from anywhere on the table. You do not have to play any other card to use the Dragon’s power (to move it & trash something).

All Keeper powers are something you can do as a Free Action on your turn without having to use another card in your had. See When a Keeper/Creeper (or Rule) says I can do something if I discard or move something…