I once had a Seneca Falls lathe and that thing would chatter if you just looked at it! However, with its major flaw, it taught me a lot about tool pressure, speeds & feeds and machine rigidity. Using a form cutter was out of the question. The closer I got to a pointy end tool bit, the better off I was. And I didn’t leave the tool bit sitting at a shoulder either!
To your chatter problem, if your carriage assembly is functioning, I would check 2 other things. (1) The chuck you are using to hold the work piece. If you can, change chucks. Like someone else has said, bell mouthed jaws or the scroll is bad. Is it a Buck chuck or like, maybe the centering adjustments are just loose on it? (2) The bearings in the headstock. Could be it has end shake play. Check the pre-load…Good Luck.