From the seal rubbing in the same spot for years it wears the shaft down. The only thing you can do to make it right is to replace the shafts or sleeve them. With the cost of that it is just better to use a thicker oil.
The snapper oil is NLGI 00; that is a consistency of about apple sauce
John Deere corn head grease is NLGI 0; that is a consistency of about brown mustard
With the heat of the gears moving it will become more fluid with use.
Either of these would work and should stop your leak. I would not go any more than these two grades in either direction because you may have leaking or not thin enough problems.
Such as NLGI 000 cooking oil to thin
NLGI 1 tomato paste to thick
For referance regular grease is NLGI 2 peanut butter