Double taxed income should only occur when you are doing two state returns, one as resident and one as nonresident. If you prepare the nonresident return first, the program will calculate and transfer the numbers for you.
If you are doing two part-year returns, again, you should do the return first for the state you moved FROM. In that case, once you allocate income correctly between the two states, you should be refunded tax you overpaid to the first state, so you would have no double taxed income for the state where you resided at year-end, the state you moved TO.
If you did the states in the wrong order, here's how to delete a state and do it again.
https://ttlc.intuit.com/questions/1900608-how-do-i-delete-my-state-return-in-turbotax-online#
