OK, at the risk of restating what may have been stated before. A little detective work:
First off, the setup for the compound and the threading tool look correct from photos 2 & 3.
Second, the OP stated that the lathe was new to him within a month but there is evidence of much use from photo #2, indicating a previous owner.
Third, From threading table in photo #4, the lathe was originally set up with an imperial lead screw.
Fourth,there appears to be an asymmetric thread as well as what looks like a double cut thread.
If the OP is using the same index mark on the thread dial, he should be cutting a proper thread, regardless of gear setup unless:
1. there is a broken shear pin (already suggested).
2. The PO had replaced the gear on the threading dial with a gear with a different tooth count.
or 3. The PO had replaced the Imperial lead screw with a metric lead screw.
I would lean toward #3 which would explain why the multiple start asymmetric threads and why it was difficult to get the half nuts to engage properly.
You can verify the lead screw by hand cranking toward the headstock and stopping when a particular index number on the thread dial is aligned with the index mark. Measure the distance from the headstock to the carriage. Now continue cranking towards the headstock until the index number is again aligned. Remeasure the distance from the headstock to the carriage. If it is an Imperial lead screw, the difference in distance will by a whole multiple of 1 inch. Most lathes are multiples of 2" but for my Grizzly 602, i6t is 3". If it converts to a nice round metric number, it is a metric lead screw.
If you have a metric lead screw, the method suggested by several above of not disengaging the half nuts but reversing the lathe to back up to make another pass will work. You should be able to cut perfect threads. They may not have the correct pitch, however. Offhand, I would not expect to be able to use the threading chart, depending upon how the change gears were set up on the lathe.