As noted by Old Salt, no need to buy a collapsible sink.  I use a small Sterilite storage box with no lid for a kitchen sink.  Something in the 10-15 qt range is perfect.  It's long enough and wide enough to allow at least partial immersion for all but my biggest cooking pot in my jumbo cook kit.

When packing up camp, I put my stoves and other potentially breakable items in this box along with small odds and ends which then goes into my gear pack.  The box protects these items and since it is full it takes up no extra room in the pack.  Also helps provide a bit of shape/structure inside the pack and keeps things organized and easy to find when unpacking.

I don't worry too much about bits of food in my dishwater.  It's kind of inevitable and if you start with hot enough water, use soap, and rinse well there's really no danger unless your dish washer does a poor job...which is going to be a problem no matter what.