How's your derailleur hanger? On a bike that old I'd be surprised if it hasn't gotten bumped somewhere along the line. That'll definitely cause the kinds of problems that you are describing.
Try shifting into a gear that makes your derailleur arm hang straight down. Now walk around to the back of your bike and see if it hangs straight from that angle. If it looks a little bent in toward the wheel, clamp a big crescent wrench onto your derailleur and bend it back straight. Don't get tempted to remove your derailleur first or you might deform the mounting hole. You should be able to eyeball it straight enough for 6-speed.