The side to side shaking I've never experienced in 18 years of ownership of a Cobra but I have heard of it on Volvo Penta SX drives, might want to see if their owners found a solution, not common but have heard of it.
The shift adjustment issues I am much more familiar with...this is at least a 3 step process and you have to do it the right way, or it will not shift the way you want. As Bruce said, you need the 3 tools (actually one you can improvise for that is measuring the shift rod height) but the other 2 are essential (bell crank holder tool and shift cable adjustment tool)
to start you remove the drive and measure the shift rod height, this must be right or else all your other adjustments will be off.
Then...the transom shift cable is disconnected both at the pivot housing and up on the engine shift bracket. The drag is measured both ways with a fish scale, no more than 2.5 lbs. Any more and you have to replace the cable. Use an OEM one if you need to do this...
Next the transom shift cable is adjusted using those 2 special tools. The idea is to get equal throw in FWD & REV on either side of neutral.
Lastly the remote cable must also be adjusted with the same principal in mind, equal throw on both sides of Neutral.
And yes the ESA system must work. It will not trigger when running on the water hose because there is no load on the prop then. In the water with a load on the prop, shifting will cause the cable sheath to flex and that movement will trigger the load lever on the shift bracket to trigger the ESA interrupt switch. This will lower the engine idle from approx. 600 to approx. 450 rpm and that allows the shift dogs to release the fwd or reverse gear. In order for it to work the micro switches must work as well as the module. Module has to have a good ground.
this is just a brief summary it is not really hard just a bit confusing the first time you do it. you do need an OMC manual and for more up dated instructions do a search on Midnight Wolf OMC parts they have updated instructions that are not in some of the shop manuals.
once it is set up right it will shift great. one finger on the control and mine goes into neutral. And it wasn't like that when I got it for sure!