I moved for a job this year and thought I was on it because I added the new state in my employers system to take state taxes out timed perfectly to the move. I inadvertently however, left my old state in too, so I was paying state taxes to 2 different states for 2 pay periods before I realized and corrected it.
Can I adjust the values in Turbo Tax for the old state that I overpayed?
(Its also a larger sum than normal taxes paid because I received my sign on bonus in one of those pay periods - not sure if this would make any difference or not)
No, you should not adjust the values shown on your W-2 when you are completing your return to account for the dates of your move. You're W-2 should be entered exactly as you have received it.
However, when you are working through the two part-year resident returns you will be able to allocate the correct amount of income to each of the states so that it will not be double-taxed. Since one state had more tax withheld that you expected, that should mean that you will just have more of a refund or less balance due for that state.