bkay - from your description it does sound a lot like a Singer 15. Do you remember if the tension pointed to the left as you sat at the machine, or if it pointed towards you? If the tension was on the nose and pointed off to the left, that's almost certainly a Singer 15. They are good solid stitchers and came in a variety of setups over the years - some are treadle, some are motorized, some have been both at some point in their history!
I have a Singer 403 and a Singer 500 and I like them both very much and have had no timing problems on either. They are slant-shank, which I think is a marvelous invention - slant shank machines are my favorite of my vintage machines to sew on. I've never sewn on or owned a 600-series Singer so I can't comment on those, but check for plastic gears. If there aren't any plastic gears, it's probably going to be a pretty decent machine.