[Al Schober]

Sally,
You say you shift manually, but it pops out of neutral? Is your external shift linkage pushing things out of neutral?
Disconnect the external linkage from the transmission lever, then try shifting the transmission just using the lever on the transmission. What happens?
The way to get the linkage and the transmission working together is not difficult, but requires a certain sequence. First step is to disconnect the linkage from the transmission and shift the transmission into fwd. Then move the linkage to the fwd position and see how the holes align. The linkage must be adjusted to give just a bit more stroke than needed going into forward. This adjustment may be as simple as a few turns on the yoke on the end of the core of the cable, or it may require moving the point where the end of the jacket is anchored near the transmission. This will get fwd where it should be. Attach the linkage to the transmission lever.
Next, reverse. First, open the transmission and loosen the reverse adjustment. Then use the linkage to shift to the full reverse position. With the linkage there, tighten the reverse adjustment until you can't turn the shaft by hand. Then move the linkage toward the neutral position and tighten the reverse adjustment another 3 flats.
Doing this should give you a good forward and a good reverse - everything in between is neutral.
In my experience, most shifting problems are due to the external linkage being adjusted improperly. Let's say both transmission and linkage want to move 4", or +/- 2". When you put the transmission in fwd, let's say that's the +2" position. If the linkage is in the +2" position, fine. If the linkage is in the +1" position, that's when problems start. It will shift fine into fwd (linkage still has another 1" to go but never gets there). But when the linkage goes the other way, problems start. Linkage goes to -2", but transmission only goes to -1" (3" from full fwd). Operator doesn't get any reverse response from the transmission, so he tightens the reverse adjustment. When he tightens enough to get reverse, there's little or no neutral - or perhaps reverse and fwd are engaged at the same time.