When I attack battlefield A with Azir, Sovereign, and the opponent reacts to the attack trigger and moves it to battlefield B, which is empty, does it retain the attacker designation and can I still move my token units to battlefield B?

Partly — your tokens still move, but Azir doesn't stay an Attacker: once the trigger resolves, "this battlefield" is evaluated from Azir's current location, so your token units go to battlefield B with him; the Attacker designation, though, is dropped for a unit that's no longer at the battlefield where the combat is happening (Rule 323.2.c).

The trigger doesn't fizzle from a move — it only whiffs if Azir is killed, bounced, or sent to Base with nothing to reference (FAQ #13069). Which tokens are eligible was locked when the trigger went on the Chain, so the same token units move; only the destination shifted.

Two follow-ons worth knowing: Azir arriving at the empty battlefield B applies Contested there, so a non-combat Showdown gets staged at B (it opens after the combat at A resolves, since only one Showdown/Combat runs at a time). Meanwhile at A you now have no units, so the opponent is the only player left there and takes control of A when that combat resolves — your tokens arriving at B can still conquer B for you afterwards.