Your next two suspects are the lead screw and the nut. You can determine the extent of each by checking backlash at the extreme end of travel.. There is essentially no wear on the lead screw there so any backlash would be caused by the nut. Measure that value. Next, m to the middle of travel and measure backlash again. The difference wold be the lead-screw wear.
Correcting each can be as simple as replacing the suspect but that assumes parts are available. Many people have made their own parts which requires a certain level of skill and equipment. A fix for a worn lead screw is to use a second nut on the lead screw, separated with a spring . This applies bias to the nut, which effectively removes backlash. This provides for any variations inlead screw wear along its length. There will still be a positional error due to the wear however.
Backlash is not too much of a problem as typical lathe use us unidirectional. Common practice is to use calipers or a micrometer to measure cutting progress, relying on the dial only for gross movements and sneaking up on the final dimension. There can be an issue with chatter because of loose components which could make parting more difficult.
