If I am filing as a resident of SC, but I moved from GA during the year, do I need to claim the GA state taxes I paid prior to moving?

Since you lived in both GA and SC during the year and earned income in both places, you should be filing a part-year resident return for both states.  That way, each state will be taxing the income earned while you lived in each location.  

The credit for taxes paid to other states would come into play only if you lived in one state while working in another.  States typically expect you to pay taxes on money earned in their state and/or money earned while you were a resident of their state.  So, if you had some overlap where you lived in SC but worked in GA, or vice versa, then that income could be double-taxed.  The credit for taxes paid to other states reduces the impact.