Yamaha 40hp - neutral position shifted

Yamaha Enduro 40hp (year unknown, it has a square cover, unlike the fancy rounded ones on the newer Yamahas).

The first time I noticed something wrong was when I put the center console in neutral, and the boat kept on going forward (it was still in gear). I disconnected the center control cables from the outboard, and found out the neutral position had shifted. If the lever on the side of the engine was straight up (neutral), the gear was in forward. Put the lever in reverse, the engine was in neutral.

I tried adjusting this, by disconnecting the shift rod from the rod coming from the gear box, putting everything in neutral, and connecting them again (found that tip somewhere on this forum). Now the neutral and forward seemed to be ok again, but shifting went rough, and reverse didn't seem to work at all.
I took it out for a few minutes yesterday to test it again, and now the neutral seems to have shifted again, and I had to hold the lever back a little to keep it in neutral...
All this time I've been testing it without the center console, so that's not causing the problem.

Something else that may have something to do with this:
The first time I started looking into this, I noticed that the shift rod was rubbing against a rubber cube, just below the engine head (top part).
I noticed there are two metal bars going into the back part of the engine, just below the head. I figure this allows the engine to tilt back from the boat, to level it at higher speeds. I put some grease on those bars, and now the engine actually slides back and forth on them... the problem is that it slides backward even when I go really slow, should there be springs or something else holding the engine, so that it only tilts back at higher speeds?
Could that be causing the problem with the gears? The shift rod does suffer a lot of abuse because of this!

Thanks!

[boats.net]

If you can move engine back and forth on transom bracket the upper motor mounts may be broke.

Ok, I found out what's causing my problem.
When I accelerate, the engine tilts backwards (see attached pictures). When this happens, I see the lever on the side of the engine changing position. When I slow down, the engine does not always come all the way back, this is why I thought I needed to adjust the neutral position, but what I really need to do is put the engine in a fixed position!

Any tips on how I can do this?
Thanks


PS I got this boat a week ago, so it may always had this problem.

[boats.net]

The motor mounts are completely shot, need to remove engine and replace mount as well as any bent linkage.