account unlock achievements across different servers

if i did the sig line on one char from AB which unlocked all the dungeons, does that achievement carry over unto another server? or do i have to re-do the sig line when i make a new toon on a different server?

It should carry over, since the dungeon unlocks are an account flag. The zone unlock for blood and sand if you did it early enough in the xpac may require a petition to acquire on an alt or redoing the first bit of the timeline(this one you can check for on your alt by doing /keys It will show up as ROR flag blood and sand dungeon unlock somewhere in the long list; sandstone delta shows up here as well).
Your alt will not be able to fast travel to the takish druid ring without doing the sig line though(or get baubles for delta without doing that line themself)