According to Le Petit Robert (dépôt légal 2001), avaler is derived from “aval” (downhill), and got the sense of swallowing only later. “aval” comes from “à” and ”val”, with ”val” in turn coming from the Latin “vallis”, which means valley.
“Avalanche”, on the other hand, is a deformation of the word "lavanche” (apparently because of “aval”), which itself comes from the Savoyard word “lavantse” and the Latin “labina”, which means collapse, landslide1.
It would thus seem that any potential common root for “avaler” and ”avalanche” is earlier than Latin.
- It’s a translation through French, so quite possibly inaccurate. The only Latin-English dictionary I found only talks of “marsh, fen”, which doesn’t match the other (Googled) sources I could find in French.