I've been trying to sell the damn Strange Amulet to Calixto so I can earn the Amulet of Necromancy like I've done in my last mage game, but he refuses to talk to me! I don't understand what's going on. Does anyone know why this would happen? He also seems to only leave his house for an hour a day around 5 p.m. then head back to his home/store and stay there all day

Here's my situation. I followed the clues to the house in Western Windhelm, and then accused the court mage (the unliving) of the murderers and then went to try to sell the amulet to Calixto, but he does not engage in dialogue with me!

Any help would be appreciated!

I have the same problem I cannot sell the amulet to him, SPOILER:
I found the guy you accuse in jail and talked to him and started a second part of the quest where I find the real killer and it is Calixto and you kill him and now I can't sell the amulet.

Same problem here. Its a reported bug on the es wiki

Yeah you have to sell him the amulet before you go and accuse the court mage, because you wont be able to after