My tax situation is slightly confusing. I work for a contract company based in California. I have been contracted out to do temp technical work for a company based in Arkansas. The contract company in California pays me to do the work. I dont get payment from the Arkansas company. But I live and or my residence is in Texas. My car is register, my drivers license, my home and my family is based in Texas. Currently I'm temporarly living out of a extended stay hotel until the job finishes out. Right now the California contract company has been taken money out of my paycheck and giving it to Arkansas state Tax. My question is do I try to go for a full refund of the amount since I'm not a resident or go for a partial refund or what. I'm really confused on this.

“Right now the California contract company has been taken money out of my paycheck and giving it to Arkansas state Tax.”----->Correct;you are NOT subject to CA income tax as you have no any CA source taxable income but need to pay tax on your income that you earned in AR to AR; AR state you work in will tax you because the income is sourced to where you work, AR.
“ My question is do I try to go for a full refund of the amount since I'm not a resident or go for a partial refund or what. I'm really confused on this.”----->Since there is NO State tax reciprocal agreement. between AR and TX, you, as a full year resident of TX, need to file your AR return on your income that you earned in AR as a nonresident of AR and also need to file your TX return as a full year resident of TX to claim your AR tax that you paid to AR on your TX return.