I WOULD do it in Riverwood, but the problem is that... everyone is trying to kill me.
Ok, I accidentally stole something. But... they won't let me pay the bounty; they're just all trying to kill me, even the civilians!
Don't know if it works, but try talking to a Whiterun guard. He should try to arrest you, pay your bounty and head to Riverwood.
Or just sneak off into Riverwood like a boss, pick up the axe, and then run like the wind.