I assume I can choose which state I vote in when I vote from abroad.
Sorry, it's not a choice. You vote in the state where you last lived before moving abroad.
As an overseas voter, you cannot simply choose any state you would like to vote in when you vote from abroad.
The last place where you lived and made a home, and intended to stay (until you left) is considered your "Voting Residence Address" or your address for voting purposes. This is the address you will use when you register and request your ballot from abroad.
If you temporarily stayed at a different address prior to moving abroad, it does not qualify as your voting residence address. A voting residence address signifies the place you lived and intended to stay and make a home prior to moving abroad.
Why all this fuss about where you vote? Because it is vital that you feel or felt "connected" to the place where your ballot will be cast and counted. Your voting residence address determines which election jurisdiction you are in, and which election office is in charge of processing your application to vote, sending you your blank ballot and counting the voted ballot that you return.
Some states allow Americans born overseas who have not established residency in the U.S. state to vote using their U.S. parent/s' voting residence address. For more information: consult the U.S. Vote Foundation Voter Help Desk to learn more about you eligibility to vote if you are a U.S. citizen born in born abroad, and which address you can use as your voting residence address.
For further information, see the Top Questions about Overseas Voting FAQ.