The copy editor is in.
I'm presenting occasional posts on the use of English,
not to be pedantic but just for the fun of language.
If you're trying to hand off an unpleasant task or an unwanted object, you're palming it off. ("I'm trying to palm off the lawn mowing onto my little brother.") I imagine it as referring to a sneaky roadside magician, or maybe a pickpocket who's trying to incriminate you — the idea is that something is concealed in the palm and then foisted onto someone else.
Pawning is the act of putting merchandise up as security for a quick loan. I'm not saying it doesn't relate at all to the way "palming off" is often used, since you very well might pawn an unwanted object, but it's not the same term.
As with many other phrases like this, though, I hear "pawn off" so much more frequently than "palm off" that I have to imagine it's only a matter of time before the former is the more accepted variation.