It is a strange thing about the feathering. I could see where if it were in the shifter it could be the safety switch. The throttle though is a bit kooky. I would take a look at the engine cut-out switch on the throttle linkage on the motor. This is the swithc that momentarily cuts voltage to allow shifting in and out of gear. The actuator on the motor should be centered except for that momentary movement just as the shifter is engaging forward or reverse. Take a look without touching anything and see if it sitting centered. If so, move the shifter in and out of gear and apply throttle. Re-center then check it again.

It may not be the problem but it's a viable culprit. I've had two "older" carburator boats that had to be re-adjusted because of this. In fact one was so bad I couldn't bring it out of gear...

I could hear the sigh of relief from here Bryan.. Now how much better does that thing run after the exhaust change etc ?