I would mark the leadscrew with a paint pen and or the handwheel and find the correlation between whats moving and the surface finish, likely something in the chicom lathe is not lined up or adjusted right and its rocking the carriage around. You can also grab the toolpost and pull back / push down while and see if the finish changes. Put a dial indicator between the carriage and one of edges of the flatways and push pull and see if something is rocking.
Properly leveling the lathe on a suitably rigid surface will return the ways in the same geometric condition they were in when it was ground. Level doesn't mean level to the earth, you just need to eliminate all the twist and a precision machinist level is the easiest tool you can easily access to do this. If do your alignment with a test part it can be hard to tell the difference between taper due to bed twist and taper from headstock misalignment. If you have a cockeyed headstock and the twist the bed to make straight parts you'll end up with the bearing surfaces only touching on the corners and it will rock and the surface finish will suck. Southbend used to only use three supports, two under the headstock and one at the tailstock, you might try loosening some of your hold downs. Your wood board might be twisting the little lathe.
It would not surprise me in the least if the contact between the carriage and the ways is poor even if its level/straight and you have what you have. Similar to how a grinder and paint can make a welder what he ain't, sometimes you end up having to use a lot of emory cloth when your equipment sucks.
This is true all true for a 8k lb lathe or a 80lb mini-lathe, the difference is there are so many other things wrong with the mini-lathes you'll never be able to tell.
Personal aside:
I have a 17x40 Whacheon in my garage that the previous shop owner said turned a taper. It was bolted to the floor and when I cut it loose to move it it jumped up. They had been using the leveling feet to try and get the taper out by pulling it to the floor. At my shop I didn't want to bolt it down as we are moving soon. Setting the 4 leveling jacks under the headstock I found that only one of the tailstock adjusters did anything. The Whacheon/Mori Seiki lathes use a huge cast iron one piece base with the lathe sitting on ground pads and bolted together. What I found is that being racking like it was caused the lathe to take a set.
Stock photo of a WL435 lathe, mine looks just the same:
Attached FileI didn't want to try and unbolt the lathe from the base as that had a lot of unknowns. Instead I made a weldment to bolt to the tailstock leveling jacks that effectively moved the rearmost leveling jack back 3ft behind the lathe. This also helped with picking it up several inches, Japanese machines are low and I'm tall. With the added mechanical advantage of the wider stance at the tailstock I could then twist the lathe back level using its own weight vs pulling down with anchors. This let me get the bed straight using a .0002/ft level. Then I found that the root of the owners problem was actually a misaligned headstock. I spun the headstock on the lathe bed using the provided adjusters and now it turns straight. I'm hoping that eventually the lathe bed assembly will flatten out over time. If it doesn't I'll bolt it down...
Don't underestimate how flopping machines are when you look sufficiently close.