Earlier quoted context omitted.
The Average Null Energy Condition (ANEC) makes it so there can not be a traversable wormhole joining two otherwise disconnected regions of spacetime. So by construction, any infinite null geodesic (a light ray) which makes it through the wormhole must be chronal (causally connected). This means there can be no traveling faster than light over long distances by going through one.
So that's it, then? Some thing, say, a million light years away from another thing is in practice entirely inaccessible to civilizations existing on that other thing, requiring a minimum of a million years to reach?
Going there and back again, and telling your pals how things are 1000LY away is likely impossible.
Expanding your civilization to more and more habitable celestial bodies, slowly but relatively surely, is entirely possible. This is how e.g. plants colonize vast spaces, being limited by the speed of their growth often by inches per lifetime. (You should also consider the amount of resources the plats dedicate to it, and their success rate.)