the black sludge is from bacteria in the pipe already starting to break down the sewage inside. The bathroom sink gets all kinds of soap and hair products, etc. in it and it connects directly to drain where the toilet hooks to. The wet vac idea may or may not work but borrow Jay J's wet vac if you do it. The sewage sludge is a mess that is hard to clean up and will really make a mess of your wet vac. The sludge is very hard to wash off your hands once you get it on them as you have probably figured out. Keep a pail under the trap arm and use a wet rag to wipe the auger as you pull it out. Avoid getting it under the cabinet and on the floor as sludge seems like it sticks like ink. Are your pipes iron or plastic?
You are probably hanging up on an elbow at 3' which requires lots of turning and pushing at the same time. With traps it is very easy to go up into the vents instead of down into the drain.
Go into the basement or under the house as well as outside the house and look for a cleanout(a wye fitting with a cap screwed into it) and try cleaning out from there going in both directions with your auger. Your small hand auger may not have a big enough tip to get an entire clog. If all else fails, get a plumber who has the proper size augers and cutting tips. Warn the plumber you used drain cleaners so he can avoid getting the caustic solutions on his hands and in his eyes, etc.