I did a little research on this, the idea thet auto unstuck will teleport you home, and it does, this is what i found..
-auto unstuck and hearthstone almost have almost nothing nothing in common, wont use hearth when auto unstucking..
-auto unstuck does 2 things, theport you home and teleport you 5 yards forward or back, the teleport you home has a 60 minute cooldown, and in between that cooldown, it moves you 5 yards forward or back(useful for passing invisible walls..)
-but if you delete your hearth(can be retreived at inn person) the only thing auto unstuck does is move you 5 yards.
* the point of this is that you have 2 hearths both on sepeate 60 minute cooldowns.
--Happy Hunting