Fausta and Faustus are much too young to hold a significant legacy (and indeed Faustus Sulla never played a very prominent role), which probably was a bonus to Sulla as he never had a clear heir that would inherit all of his political power while Dictator (he had supporters of varied loyalty like Lucullus, Pompeius, Metellus Pius, Vatia and so on, but considering how the 70's developed they don't count for this excercise).
I don't think the scenario plausible given the attitudes held towards Sulla and the fact that he never had that much of an actual powerbase (his veterans were ailing, and Sulla was never the predetermined champion of the Optimates, it looks more like an alliance of convenience), but if a POD is needed for Sulla to have a strong somewhat adult heir you could have his son Lucius Cornelius Sulla (his mother being "Ilia" or a Julia Caesaris) living to adult age. Assuming his son to have been born between 110 and 100 BC, he would be between 20 and 30 by Sulla's death, somewhat like Marius the Younger.
But of course, his survival could also affect Sulla's career in a significant way, given how unlikely his rise to power and prominence was.