If Albucierre drives exist, does this imply that communications can travel no faster than the Albucierre driven vessels?
Yes. But that doesn't actually mean anything.
Because Alcubierre drives can travel through time.
If the only technology in your setting not backed by modern hard science is the Alcubierre drive, then there is no other technology in your setting for FTL communication, by definition. You could easily add one, of course, but that's getting off-topic.
Any FTL travel technology can send matter and information backward in time if and only if it is capable of making FTL jumps in different directions in different frames of reference moving relative to each other. Alcubierre drives are actually one of the hypothetical FTL drives most amenable to exploitation in this manner, since the simplest way for them to work is for each FTL jump to take place in the ship's proper frame (that is, the reference frame moving with the same velocity as the ship, in which the ship is stationary) before the jump.
Thus, a ship could fire up its Alcubierre drive, zip off into interstellar space, drop out of warp (moving at the same speed it was before the jump), fire up its conventional thrusters, accelerate away from Earth to nearly the speed of light, hit the Alcubierre drive again, and return to Earth several years before it departed. Because special relativity.
That explanation made no sense at all, so here's a video from PBS Space Time that explains the FTL time travel process much more clearly.
I will note that, with this system, it is much easier to send small packages back in time, since that crucial conventional-thruster step in the middle will require burning a huge amount of propellant. Your engineers may find that sending people back in time by this method is economically infeasible, but sending a couple of SSDs in a tiny Alcubierre-enabled ship could be profitable.
If you want to forbid FTL time travel, you need to ensure that all FTL travel occurs in the same universally agreed-upon frame of reference. This is probably easier to justify with a technology that teleports ships instantaneously from one place to another, a lá the Teraport in Schlock Mercenary. Just declare that all teleportations are instantaneous in the same specific frame of reference (e.g. the frame of the cosmic microwave background), and you're good to go.
If you're OK with FTL time travel, well... go wild.