I know this is a wild theory here, but it also fits two recurring themes of Single Again:
What if Trevor, Shannon's first husband, isn't Lilly's biological father?
We can already rule out Becca being his daughter, since Shannon would have had at most a month to finish high school, find a job, and get to know Trevor. Theoretically possible, but practically very narrow.
Accordingly, Becca and Abby definitely grew up completely without a biological father. With Becca I assume it was a drunken one night stand at a girls night with Shannon and her friends, how it was with Abby or rather her mother we know.
So what would the scenario look like? Shannon and Trevor had been married for a few years, about 3-4 years maximum. Shannon might be bored and Trevor might not pay enough attention to her to satisfy her appetite. So she claims to be visiting friends (such as her high school clique or other friends she has since made) and either assigns Trevor to babysit Becca or pays a babysitter. She in the meantime moves around the houses, she is after all only around the 20 and from therefore this is perhaps not completely unexpected, meets a guy she likes, one has a few drinks too much and bang they end up together in bed, as already with Becca's conception.
So Shannon gets pregnant and either doesn't know herself that Trevor is not the father or hides it from him. Lilly is born a few months later and of course thinks Trevor is her father and is totally fixated on him as a daddy's child. Trevor eventually realizes that he is not the father, either relatively early and pulls himself together or by a stupid coincidence. In any case, at some point it becomes too much for him and he leaves and Lilly's heart breaks.
Of course, it's also possible that Trevor is actually Lilly's father and he's just a bad person and father. That's also possible, and bad parents, especially bad fathers, are also a recurring theme in Single Again.
