I haven't bought a chopper in a couple of years, so I can not speak to the foam issue. Lots of other good things to try, though. I did move my hinge supports in to take out most of the side play, like has been mentioned. What most helped one of them was shimming the hinge support. I looked at the chopper alignment on one (without a feather in place) that was requiring two smacks. The blade only touched the striking plate on one end. That would have to place undue stress on the the hinge and support area to get it fully down, let alone smacking it twice.
I used some card stock (like from a stiff, manila folder or 3 x 5 card) to shim up the low hinge side. The cutter on the plate surface was much better and took only one smack to make a nice cut. Just one more thing to look at.