Purchased a one story house on a basement, built in late 70s. When we first moved in the washer would drain into the utility sink (beside it), and then excess would continue to drain come up in another sink in the next room (the wet bar). This occurred every time the washer was used. Now when we use the washer or dishwasher (located on 1st floor) the floor drain in the utility room backs up and floods. The smell is awful, and it emits black, soot-filled water. What could the source or problem be? Snaking the main floor drain has been unsuccessful.

That's not soot, you know that , right? Your main drain line under the floor is clogging up and needs to be rodded. Any clean outs on the main verticle stack?

Yes, I know that is isn't soot, but am trying to avoid reality. I don't think much of anything has been cleaned out, not sure about before we moved in... from the way it looks they haven't had attention in a while.

You have a clog downstream from the floor drain that must be snaked out. This can be reached from the utility sink roof vent or a cleanout located on the stack.
If the floor drain has a plug located under the cover you can snake from that. Good luck, Tom