The kitchen and bathroom share a wall. The drain beneath the kitchen
sink (double sink; one side with disposal and dishwasher) goes into
the wall it shares with the shower. When we run the sink (especially
with the disposal on), the filthy water comes up from the drain in the
tub.
The tub drains mostly fine, though it has had it's problems in the
past. But the dishwasher doesn't clean as well as it should, and I
wonder if it's backing up inside.
I have tried running a hand-crank snake down the tub, but this hasn't
helped much except to make the tub drain better.

If you've already cleaned the drains, I'd take a look at the roof vent.
Since the kitchen and bath share a wall, I'm guessing they share a roof
vent too. If the vent is plugged, none of the fixtures are going to drain
all that well.

Anthony