I just purchased a new (well, new to me) quicksilver 3000 classic throttle for replace my old throttle.
When I push in the "throttle only" button and move the throttle forward, should the shift cable still be moving as well??
No. Pushing in that button should disable the mechanism to it. A shot of lubricant through the brass handle retaining nut often frees things up. I use white lithium spray grease.
To pop the Throttle-Only button out, you need a tiny screwdriver to reach under and pop it out. Unfortunately the control has to be in Throttle-Only mode with the throttle advanced forward or it won't pop out. The hole that the buttom passes through is where to shoot the grease.
Make sure it's actually in neutral when you try to use the throttle only button.
Had a guy bring one in to the shop because he couldn't get it into throttle only. He moved the lever to where it was straight up on the unit and looked like it's where neutral should be, but it was actually in forward. The handle wasn't on straight with the unit.
