Why is Rosa embarrassed? They're quite open about being in a relationship; the whole of Baron already knows (and if this is at some point during the game, as evidenced by Rosa being young, Cecil being a Paladin, and Kain not meditating on Mt. Ordeals), some players theorize she may already be in the first few weeks of being pregnant with Ceodore.
It's a little late to be hiding your face, Rosa, is what I'm saying.
At the very beginning, Cecil and Rosa are in a "but thou musn't!" loop, due to class relations or somesuch (at the least, him being a dark knight). Since he's a paladin here, though (with built-in scarf!), it's pretty much past the point of denial.
Maybe she just wants to keep up some sorta front. Not that it'd do much good; even the castle's white mages have discerned enough to tell Cecil what-for before you even leave the castle.