For WSNM trials, you talk to the NPC and have the option to give up on the WS--then you trade the weapon back if I remember right. If you chunked the weapon, then you may be in a pickle and need a GM to assist you.

Not sure how the mythic line goes...never got around to mine. Logged my club, then stored it since I'm not done with Nyzul yet--static broke up ages ago and just never got back to it.

From the wiki:
You can have multiple versions of the quest open at the same time.
So, I guess there isn't anything blocking it--might need to equip the new weapon and talk to the guy in Jueno again, idk.

As ffor Blood and Glory, I think the only thing other than another WSNM trial being open, I think the only thing that would block you would be the Curses... questlines that lead up to Warp2, or maybe the MKD one. If one of these is still open, need to finish it so she will log it. The Errand one for the professor isn't supposed to block some other quests though, so it may not be affecting the Staff WSNM. Here is a link to Shantotto--might want to check your logs to see if one of these is open, and try closing them out to see if it fixes it:
http://wiki.ffxiclopedia.org/wiki/Shantotto