welcome to the forums. chances are ?, your shifting shaft needs replacing ?. without having my hands on it ?, I can't say 100% this is your problem ?. there is no way of knowing for sure , until you did deeper into the motor .
Yep, if the shifter is able to be moved around then I would guess the shift shaft is broken, and sounds like it would be at the back. You'll have to pull the rear cover off to see. If it is broken, you'll have to pull the front cover as well to put a new shift shaft in.