MFS5B 5HP 4-stroke jumps into gear when the throttle is opened

If I race the engine while it's in neutral, sometimes it kicks into gear.
Shall I change the gear oil?
Or, is there a shift-shaft adjustment of some sort?

The engine has very low hours on it.

Sounds like either the shift linkage is maladjusted, or there are broken/missing parts. Change gear oil and motor oil annually or every 50 hours, whichever comes first. Never race the motor in N. When did this start? Has any work been done to the motor recently?

Well, I decided to change the gear oil. Judging from what came out,
it was very low...but, no water.

It has been this way ever since we got the engine.....about a year ago.
I'm pretty sure the prior owner did not take it in for service.

I've noticed that it is not very clear when it is in neutral.
It doesn't really "snap" into neutral. Enough vibration and
it drops into forward.

Is there a shift-shaft adjustment under the "rubber disk" on the driveshaft housing?
I do not have a Factory Service Manual.
Prior owner might have messed with it.....

The LU only holds about 195 mL of 90 weight. Be sure you have good fiber washers under the heads of the fill and drain screws. I am sure the LU is not taking itself into gear, unless the clutch spring is damaged. The shifting system has a ceramic ball that causes the shift handle to click into F, N, and R. There is a brass coupler between the shift rod and the cam rod. That could me mis-adjusted, but if so, you usually have only one gear (say, only forward for example). See the diagram:
+
Dunno why the previous posted diagrams got into trouble. Sigh.

I took the rubber plug out.
The head on the bolt that goes through the brass coupler is in the middle of the access hole when in reverse.
In neutral, it is at the top of the access hole. In forward, it is out of sight (above).

The coupler is right against the stopper on the shift rod.
Perhaps the cam rod is not in the right spot.

Is there a simple adjustment procedure, or do I need a Factory Service Manual?

You can unlock the coupler. Make sure you know which rear the LU is in. I would suggest Reverse. Do not remove the bolt, or loosen it more than a half turn. Then cycle the shift lever. You should feel noticeable detents. If not, the ceramic ball and spring may need attention. Shift the lever to R. Then tighten the coupler. Done.

So, I reset the cam rod by loosening and re-tightening the brass coupler.
Dribbled some WD-40 down the cam rod towards the cam rod bushing.
And, cycled the shift lever quite a few times with the engine horizontal (oil inside the gear case).

It shifts quite crisply, now.
Although, I have not started the engine, yet.

One thing I did notice is that the engagement of the "starter lock cam" with the "starter lock"
(up at the top of the recoil starter) seems a bit off. See the attached photo (engine in neutral).
The part number for the "starter lock" is 3R1052520. Apparently, it is an "OLD" part.
Has there been a redesign of this mechanism? If so, what are the new part numbers?

(Couldn't get the photo uploaded...will try again later).

There have been a couple of redesigns. No biggie. Bend the link rod until when you are in N, the lockout is at the top.

When adjusting these shifters, I usually get the LU in a known gear (or neutral), unlock the coupler, and then set the lever in the same gear, then lock the coupler. All your WD40 did was make the top of the LU wet.