The lead screw is usually not a single continuous rod (which would be too large to package and ship). My Genie is broken into three segments, and the joints have a gap with no thread. That's normal.
The carriage seems like a good place to start. I'd also make sure that everything (screw, carriage, door rollers, spring, etc) is properly lubricated, as friction can cause problems. It's very easy -- just make sure you use lithium grease and NOT WD-40.