Hi,
As the backlash & free play don't seem to be the issue, it is time to get back to the basics for a while. I started thinking what could cause longitudinal cuts to be bad but not affect facing cuts. Both of these suggestions are really "back to the basics"; I am asking them since we don't know your background and it is possible that you have overlooked these. Please don't be offended if these are already crystal clear to you :-)
1)
Is your cutting tool set at exactly center height? If the tip of your cutting tool is below or above center height, your facing cut could still be ok but the longitudinal cut would not.
Put a dead center on the headstock (or tailstock, assuming your tailstock is properly at the same height as your headstock; it should be...) and compare the height of the point of the dead center to the tip of the cutting tool. Are you below or above?
Or you can test it like this (assuming your tool height doesn't change when you change between facing and turning): make another facing cut. When you are done, is there a small nib or pip left at the center of the part? If you see a nib, you are below centerline. If you don't see a nib, you could be above centerline. Adjust the cutting tool down and make another facing cut; repeat this cycle until the nib appears and then go up just enough to get rid of it (or until the nib that is left is so small that you can pry it off with your fingernail).
Here is one way to check the center height of the cutting tool:
http://www.instructables.com/id/Correct-Lathe-Cutter-Setup/ (Click on the "View all steps" button).
2)
How are you feeding the tool when turning? Are you feeding it manually or using a power feed? If manually, have you noticed that the "gearing" of the handwheel on the saddle is way faster than on the crosslide handwheel and you should turn the wheel much slower. If using a power feed, what is your step per revolution (0.1-0.2mm / rev aka 0.005-0.01" / rev is in the ballpark). If you are feeding too fast compared to the speed at which the part is turning, you will get the screw thread effect.
If neither of these was it, then I think some pictures would be good. At least a pic of the finish that you get when turning would be needed. Seeing what kind of screw thread effect you are getting will probably be the key. A pic showing the cutting tool near the turning part might also be helpful.
Cheers,
Jari