If you list yourself as a full-year resident of IN, IN would then tax ALL your income, including income from other states.

Since you moved in 2019, you file as a part-year resident of IN, and a part-year resident of your previous state (assuming your previous state has an income tax).

In the personal info section of TT, indicate that you lived in another state.  You became a legal resident of IN on the day you began living in your new home in IN.

The issue is that TurboTax is not recognizing the county tax that I owe. By choosing that I was NOT a resident of an Indiana county on January 1, 2019, which is true, it's stating that my part-time residency for the year is resulting in a $3,008 refund from the State rather than owing $5. County tax has been withheld from paystubs so I'm not understanding the discrepancy.

According to the web reference below, if you are not working in or a resident of Indiana on January 1st, you are not liable for county tax for the entire year.  

"If an employee both lives and works outside Indiana on New Year’s Day, they are not subject to county tax for the entire year even if they move to an Indiana county on January 2."

https://accupay.com/indiana-county-income-taxes/

So that $3,008 is likely a refund of county tax that was incorrectly withheld from your pay.