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Have you checked to see if the upper linkage is worn out.
The motors old and up in the lower cowl the upper shift shaft (linkage) has a couple of "latches" and at least one wear plate (all small spring type steel) that kinda (clicks) the linkage in/out of gear when you shift (I'm just talking about the linkage end here, not the gears controlled by the lower shift shaft).
Anyhow, if one of these latches is broken or worn, or the wear plate is worn, a bit of torque in either forward or reverse (respectively) can be enough to "kick" it out of gear.
Quite a few merc's have a little cam on the upper shift shaft and the spring steel latch on them does fail as they get old, resulting in the same symptoms.
Your set-up is a touch more complicated than the simple cam, which is why I would at least take a look at it.
With the motor in gear (reverse since it's your problem at the moment), but the motor "not running", and then try Jeff's suggestion of grabbing the prop - torque it (by hand) back and forth a bit an see if you can get it to pop out of gear - if you can, get an "assistant" to hold the linkage under the hood and try again...
This aint good as you could have damaged the shift rod ...I had to tighten the bolts to draw the lower unit to the exhaust housing. I figured that was because the water pickup sleeve was messed up.
On this model should be about .050.Grab the prop and see how much fore and aft motion the shaft will allow. If more than a few thousandths, that is too much.