When the new Deadman from Trinity is assigned a relic from a resource and he is hit to his countdown clicks and moved to his card, what happens to the relic/resource fragment? Does it stay on him? does it drop in the square he was occupying?

He's not ko'd, so as far as I know the relic would remain assigned to him and on his card.

This is 100% accurate. He is not KO'd/Defeated so the Relic stays with him.

"I'll just take this hammer back to the afterlife with me"

"This guy has a hammer!"