Oh man - I would absolutely keep my mouth shut.

Maybe the guy at the check station thought that you shot it the night before, and just got in to check it in that morning??? Don't you some time to go and check it in? 24 hours maybe?

You realized you made the mistake. You didn't mean to do it, but I don't see that as a reason to rush off and turn yourself in. The words "Don't beat yourself up over it" comes to my mind. You made a mistake. You didnt mean to do it, and now you seem to me to be sorry about it. Seems the fear of it all might have scared you enough to prevent it again.

Call in - I would think that you are guarenteed to pay the fine.
Don't call in - and you only MIGHT have to pay.

Everytime you accidentally speed, do you stop at the first policeman you see and ask for your speeding ticket?

I know what I would do.
1 - Be more aware of the laws
2 - Hush.