Perhaps this story was later, or earlier, called "The Tau Factor".
It was about a space ship full of colonists headed for a planet orbiting another star. The ship reaches near light speed by using magnetic scoops to gather in interstellar hydrogen and put it through a fusion reactor. The scoops also protects the crew and passengers from radiation which would result from collision at relativistic speeds with the interstellar gas atoms.
A collision with an unexpected interstellar dust cloud damages the drive so the crew can't shut it off to repair it, or reverse it for deceleration, without also shutting off the protective shield, though they still have some steering control. They continue to accelerate closer to the speed of light, with inherent time contraction, leaving the galaxy, then the local galactic cluster, to find an area where the hydrogen is less dense. They choose a trajectory that will later take them into another cluster.
By the time they get the drive repaired, studies have shown that so much time has passed outside of the spaceship that the universe has stopped expanding and is starting to contract. They aren't enthusiastic about colonizing a dying universe, so they go into orbit around the "big crunch" and colonize a young second- generation solar system in the next expansion.
I suppose the obvious problem here is that they couldn't find any way out of the universe to avoid the collapse. This may still depend on how the universe is really "shaped". Are there any other serious difficulties with the story?