Both are certainly possible, though I do agree with Blais here that it is strange that everyone died except Littlefoot's mother and grandparents (and I guess Bron too).
Since the narrator said that some herds stopped to hatch their young, I suppose it's possible that the longneck herd carried on while the family stayed to watch over the eggs. It never actually shows any longneck herds besides Littlefoot's family.
Anyway, I've looked through a few things, and this is something I came up with:
I saw "Legend of the Story Speakers" today, and I think Saro may have been part of the herd, before being separated somehow. Since Grandpa says something along the lines of "Saro? Is that you? I had almost given up hope!" <--- Did Grandpa think that Saro may have died?
Another thing that I don't quite get is when Bron says that there was a huge crack in the earth when he got to his home. From what I gathered from the first movie, Littlefoot's mother and grandparents were already on the journey to the Great Valley, and a fair bit of time seems to have passed between Littlefoot's birth and the next scene, where it is shown that they are travelling. Could this mean that there was perhaps a different earthquake which caused that crack, possibly separating the herd? I'd say that Littlefoot's family were close to the the front of the herd and managed to escape, some in the middle fell to their deaths, and those at the back, possibly including Saro, were stuck on the other side. After that, some probably died of starvation and disease, while others joined other passing longneck herds for protection. Perhaps a few of them even found Bron and joined him. If there were any others on the same side as Littlefoot's family, then, as Blais said, they probably carried on when Littlefoot's mother laid her eggs. Saro, either waiting until the Great Drought had passed, or when he realised that everyone was forgetting the longneck stories, later set out to find the Great Valley and reunite with his old friend again.
I just made this up today, so I wouldn't be surprised if there were a load of mistakes and things that don't make sense in this. Well, nobody knows what really happened, but I really do like to see people's idea and theories.