It is possible that you have a frozen bellcrank in the pivot housing, but there could also be a problem in the remote contol unit. The remote control has 2 cables that go to the engine. One connects to the throttle lever on the carb, the other connects to the engine bellcrank lever. The way its supposed to work, is that the control shifts the drive into gear, and then advances the throttle after the shift was completed. Have someone watch both cables when you try to shift it (engine running on the water muffs). Does the cable that's connected to the bellcrank (slotted lever) move when shifting and is the transom shift cable moving as well? If none of it is moving, then I bet the problem is with the control or cable. If the transom shift cable is straining to move, its the pivot bellcrank, or a rusted internally cable, or something in the drive.
While many will tell you to get rid of it, these problems are usually something simple. The cable broke, internally corroded, the pivot bellcrank got salt water deposits behind it and froze up (rare but can happen)
etc. If it was shifting well before, think about what could have happened over time. Corrosion.
The Cobra system, when you know how to set it up right is actually very good. It takes patience and time to understand how its supposed to work and to understand why its not working.
I would get those tools, shop manual instructions etc off that website.
I'd pull the drive off and disconnect the cable retainer on the bellcrank lever and and the engine end to start. Measure the cable drag with a fish scale. If its more than 2 lbs, then you need a new cable. If not then something else happened.
Also, you should be able to shift the drive into FWD, N and REV by using the shift rod, while someone rotates the driveshaft. While you have it off measure the shift rod height. If you have a good cable, all 3 adjustments are right and the ESA works, it should shift fine.