The maintenance of public records, like marriage certificates and property records, would fall under the jurisdiction of what executive officer?

The executive officer would be a County Clerk.

The County Clerk is a Constitutional officer who is elected every four years. He is responsible for keeping and maintaining records of births, deaths, assumed names etc. He is issuing and filing marriage licences, gun permits and notary bonds. He is also responsible for processing passports.

County Clerks can either be elected or appointed, depending on a particular state's law.

in about winter like late november to january