cuisiner des pâtes au poulet et dessert au chocolat

What does “je suis débordé” mean in French?

Déborder is what a pan does when you fill it too near the rim and the water boils over: literally to go past the edge, le bord. That picture is why je suis débordé(e) is the ordinary French for “I’m snowed under” — more is coming at you than the pot can hold. It is a normal thing to say at work or to explain a late reply, with no drama attached.

The same verb also covers things spilling over in a good way, with de: déborder de joie, déborder d’énergie — brimming rather than drowning. The preposition is the clue: débordé alone is overload, déborder de something is abundance.