You've kind of nailed it down right there.
I think you ONLY have a leadscrew, no power shaft, but it doesn't matter. Whatever shaft you're driving from, if it does not stop, but the carriage does stop... Then it's inside the apron. Or behind the apron, depending on how you consider the rack gear and pinion,...
Something's loose, something's broken, something's worn, or something's out of adjustment. You might have some tactile feedback when it happens, or maybe not, which could give you "some" clues as to the nature of the failure... The problem could be several things, but you already know where you're going to find it.
The only thing I can think of that you might do before you take it to bits is to consider the clutch/engagement mechanism for the power feed. I do not know the answer, but if there is an adjustment there, a friction clutch or a sliding gear, or heck, even a loose pin in a lever, making a poor engagement, that "might" be addressable without disassembly. Have a look at that of course, but I think you're goin' into that apron.