Not sure what you are looking for exactly but this is what I searched for. I looked for the deaths of his parents but only found one Cox listed in the timeframe you stated. It was a William Rousseau Cox who died in Natchez in 1832. Doesn't sound right. Then I searched for a marriage for William although I know you said he went alone to MO. As expected, I didn't find one.
He didn't spend much time in KY. Sorta passed through. He was very young to be doing all this on his own, but not unheard of in those days.
Most of the death records I have are later than the 1830's. i have a few court records but only for Estill co.
As I said before, I wasn't sure what you were looking for exactly. If you are looking for the location of the deaths of his parents, I would start with the counties nearest to the boarder of Ohio. You might do some research on the migratory routes in the 1830s. One guess is that he traveled down the Ohio river via packet or flatboat to the Miss. River and then to St. Louis. This was a fairly common way to travel. My own gg grandfather was a riverboat pilot on the Kentucky and Ohio rivers.
Good Luck in your search and I'm sorry I wasn't able to help you out.