I had the shock/damper fail this fall on my GX345 and the symptoms were a little different than you described. In my case, the pedal would stick in position and not return to neutral - I almost careened into my car while parking in the garage when it first happened. From my understanding and what the article posted by Oldgibsonguy indicates is that the shock is to slow the return of the linkage to neutral when you take your foot off the pedal to prevent the tractor from abruptly stopping, not to slow the starting.
It dampens the movement of the linkage but you can easily overpower when you step on the pedal. Any chance the pedals are returning to neutral slowly giving the illusion that the transmission is hesitating for a second? Does the pedal feel normal? Mine took significantly more effort to depress.
You should be able to disconnect the damper and drive the tractor with it disconnected, just try to slowly release the pedal if you disconnect the damper. Also, if the damper is leaking fluid, it needs to be replaced.
if you do need a damper, they are a little cheaper (at least for my GX345) from TT than from JD and with some internet sleuthing, you might be able to find the OEM part online for much cheaper.