I guess yours is the 3" Special as it has a single sided feed nut. The 4" Special I had many years back had a proper two part nut which, as I understand things, was the case with all the Pools lathes except the 3".
The 3" Special is actually a re-badged RandA so searching under that name may turn up more info. Normally the single sided feed nut configuration has a fairly long, almost half round, shoe beneath and close to the leadscrew to support it against the pressure keeping the nut engaged. This system can work fine, Colchester and Kerry were two makes that I know from experience made effective use of it. Theoretically the shoe can collect oil helping to keep the screw lubricated. It can also collect debris helping to wear it out. Given the sketchy, open, nature of the apron I'd venture to suggest that the latter is the case with your machine. Easily remidied by making a new one. This machine requires the leadscrew to do double duty as a feed rack in addition to its normal job. The feed pinion causes extra wear which not only reduces the engagement between leadscrew and nut but also promotes more rapid wear of the nut.
Clive