If Azir Sovereign moves to a BF and drags a SS token with it and the token was ready before moving, is the token exhausted after all said and done? Or no?

No, the token is not exhausted after the move. It stays ready.

Azir's ability is a triggered ability that moves token units — it is not a Standard Move. Standard Moves require exhausting as a cost (Rule 144.2), but spells and abilities that move units do not change the unit's exhaust/ready state unless they specifically say so (as established by FAQ #9376 / #7964 and echoed in FAQ #1531, which confirms Azir can even move already-exhausted tokens). The token lands at Azir's battlefield in whatever state it was before — ready stays ready.