Assuming the throttle is opening all the way, is the carb to magneto “Sync” okay?

I’m sure someone else will have some better ideas.

Well,  I’m sorry you swapped out the gearcase.  Are you sure the prop is OK?  How about the shear pin?  Perhaps the engine is mounted too high on the boat or trimmed out too far.  Perhaps you have a short shaft engine on a long shaft transom.

OK, so you have the other gearcase on now.  Recheck the shift rod connection, be sure the clamp screws are properly located and snugged down tight.  The threads on these screws strip out easily, so be sure that is not the case.  Once  proper shift rod connection is confirmed, then go ahead and adjust that fine adjustment on the shift lever.  Have you ever used this gearcase before?  It is possible that it has clutch dog issues, once the clutch dog/gear lugs are rounded no adjustments can repair this, the gearcase has to come apart.

So, the issue seems to be a little confused now….  Does the second gearcase misbehave just the same as the original?  You mention “jumping out of gear”, there are two different symptoms that can be occuring here.  A slipping prop, or ventilation issue will show symptoms like a manual shift car with a bad clutch.  The engine will rev up when the slipping occurs without really pushing the boat.  Jumping out of gear has a different type of symptom.  The engine will “jump” or “buck” when the clutch dog/gear lugs bounce over each other under load, almost as if you were hitting something in the water or the engine had an ignition miss.  A partially broken shear pin could exhibit these type of running issues as well.   The possibility exists that your original gearcase was alright, but you had a slipping prop or ventialtion issue, perhaps a partially broken shear pin, and your second gearcase has clutch dog issues.

Do you have a different prop to try? A new shear pin?  I’m hoping the issue is a simple as one of these things, but fear your second gearcase has issues.  So, perhaps you can elaborate on the symptoms on both gearcases.  A video of the engine misbehaving would be a big help as well.