Brief update on this thread: After contacting a pro drain cleaner and waiting for response I decided to try and auger the drain myself at the trap (I'd already passed a snake through the cleanout but it did nothing.) The other guy who came by said that he couldn't get the line past the trap.
I only have a 50 foot 1/4" auger .. never felt like I "passed" anything but it kept taking the line slowly, about 12 or 15 feet. Brought up some debris but I suspect most of the blockage is rust and the like. Anyway, when I tried the drain again with the garden hose it seemed to drain better and when I ran a load of wash it did not overflow for the first time in weeks.
I suspect it still needs a more thorough cleaning and that the problem will come back . but at least I know that the issue is at or near the trap (and thanks to the advice here I did not go forward with getting a sink in the room.)
