If the Septim line of Emperors had not ended with Martin during the Oblivion crisis, would they have been able to kill of the dragons and absorb their souls?
Since there were no dragons alive during the Septim dynasty, it's unclear how much Dragonborn was left in the Emperors. I find it unlikely that Martin's descendants would have been able to absorb souls and use the Thu'um, though.
What Makes A Dragonborn?
For starters, we don't really know how a "true" Dragonborn is born (in the sense of someone that can absorb souls -- there have been "Dragornborn" like Alessia that couldn't). There have been extremely few Dragonborn in history, and rarely more than one at a time. If it were purely genetic, you'd expect to see whole families of them, which we don't.
We also don't know how "diluted" the Dragonborn trait might become over generations. Martin was something like 15 generations removed from Tiber Septim, so who knows if he was even the tiniest bit Dragonborn.
There's also the possibility that something more metaphysical is needed to be a Dragonborn. Over the intervening centuries, there were some pretty weak and even a few crazy Septims. Those don't sound like the traits of a Dragonborn, so it's possible they were no longer "worthy" of it.
Who's Really A Septim?
But even if we assume that it's completely genetic, and thus Tiber Septim's children would inherit it, there's the problem that the Septim dynasty were not Tiber Septim's direct descendants. The only immediate descendant of Tiber Septim that ruled was his grandson Pelagius I. Tiber did not have any other children or grandchildren, so when Pelagius died, his niece Kintyra took the throne. So, from the very start, it's not clear how much of Tiber's "bloodline" was present in the Septims.
The question becomes even murkier during the infamous reign of Katariah and her sons. Katariah had two children: Cassynder by her husband Pelagius II, and Uriel by her consort. Upon her death, Cassynder was declared the "only living blood relative" of Tiber Septim, and thus pressured into taking the throne. He subsequently adopted his half-brother into the Septim family as his heir.
When Uriel died, his "cousin" Cephorus was put on the throne instead of Uriel's son, but I can't find any record of exactly how Cepohrus was related to Tiber Septim. If Pelagius II had any brothers or sisters, it wouldn't make sense to call Cassynder the "last" descendant of Tiber Septim, but Katariah's brothers or sisters wouldn't be Septim at all (she was a Dunmer). At that point, the Elder Council was calling the shots, so Cephorus' "claim" to the throne could easily be political: a link so minute it barely even counts, but enough to legitimize his claim against Uriel's son.
With all of that confusion and uncertainty, it's difficult to imagine that every Septim emperor was automatically a Dragonborn. More likely they just claimed that title because, with no dragons around, no one could argue with them. But unfortunately there's literally no way to know for sure.