.....it took nearly 5 months to close, and we didn't try to rent the property during that time. Does that mean I should pro-rate things like the property tax and insurance payments to only include those months that we had tenants?
Yes. During the months in the 2019 tax year that you had tenants, property taxes and mortgage interest would be entered on Schedule E.
If the property was not held out and available for rent during the time it was for sale, and you itemize deductions, then you would enter the mortgage interest and property taxes (subject to limitations), during that time period, on Schedule A.