Obviously needs a good cleaning & lubrication. Today's marine grease (Triple Guard) will not dry out like that. The part number you seek is #310700 (I think you found it), but Crowley Marine has it for $17.99 & less shipping ($3.95 I think). eBay only showed 1 for $25.95 Free Shipping.
The key here is when you install it (all lubed up & moving freely) before you hook the cable to the engine to make sure everything is working well. After you hook up the cable then make sure when you return to idle there is no preload on the cable. You just want the cable to return the motor to the idle stop,on the motor. Either with the throttle handle or the idle adjustment lever you just installed. You may have to adjust the trunion at the motor to achieve this? If you preload the cable then the situation you complained about will happen. When you shift (throttle up, shifting is just a switch) the warm up lever will move. Any binding of the cable due to preload is very hard on that little lever!
A separate switch for shifting is not a good idea. It has been done because the switch is expensive or NLA, but shifting at higher than dead idle is hard on the lower unit components. IMHO
Dan in TN