My residence is in Texas. I work in and live in Texarkana AR Monday through Friday. Do I file non-resident status, or part-year resident stat us?

If you stay in a hotel Monday thru Friday then you could make a case for part year residency.  If you maintain a residence (bought a house or rent an apartment) then you are a full-year resident of Arkansas.  Either way you are exempt from the Arkansas income tax for money earned while you are a resident of Texarkana even though you are required to file an Arkansas tax return so I would just put down that you're a full-year resident.  If you put down that your primary residence is in Texas they have to start withholding again even if you're exempt from the taxes.

Here's the letter related to the exemption.

@child-o-the-King