It sounds like your ship has to use Dead reckoning. When there is too low of visibility, pilots and ships captains will use the knowledge of where they were combined with speed and time to determine where they are.
Since you'd be relying on dead reckoning to determine the position of the destination AND the position of the ship it is absolutely vital that:
The measurements of the stars are ABSOLUTELY accurate. Since you're extrapolating over billions of years the tiniest error will get your crew lost in space.
The consistency of the warp drives movement. Again, this most be ABSOLUTELY consistent in order for your dead reckoning to be accurate.
In terms of world building this would mean that the hundreds of stops are at closer planets in order to verify that dead reckoning is successful. The Alpha Centauri system is about 4.37 light years away, so it could be used to ensure that the dead reckoning is accurate, from there your crew could just warp exponentially further each time while constantly checking to make sure they're ending up in their expected destinations.
The big problem: It is impossible for us to know how accurate our measurements of stars are Since we won't live long enough to see the results of the experiment, all our hypotheses about the direction and speed of a galaxy are educated guesses that can never be proven or disproven. I can think of some ways around this:
Perhaps the ship your crew is on is the first ship to travel long distances and they test the accuracy of the measurements using the process described above.
Perhaps dead reckoning has already been tested by other ships or space probes using the warp drive and the process is commonplace in the future.
If this is REALLY far in the future you could say that they found our current day star charts and by comparing them to their they could accurately extrapolate out the movement.
Overall it would be a process of trial and error, but given the invention of a warp drive I see the navigation method you're describing as a perfectly valid method of space navigation.
EDIT: It seems like the main question is the precision of the warp engine. If it can instantly get you where you want to go every time you don't need navigation. If it's inconsistent you should explain more about why it is inconsistent and how inconsistent it is.