No, you cannot choose the state where you will vote. U.S. citizens living outside of the U.S. are only permitted to register and vote in the state and county where they last established residence (domicile) in the U.S. before moving outside of the country.
To register to vote and request an absentee ballot as an overseas voter, you must submit the Voter Registration/Absentee Ballot Request form (also known as the Federal Post Card Application, or FPCA). You can generate the completed form on our website.
(If you are serving in the uniformed services, or are the spouse or dependent of someone serving in the military, you can find information under this question: "As a uniformed services member, spouse or dependent, how do I determine my legal residence address for voting purposes?")
You will use the address of the last real home you had in the U.S.—where you actually resided—and that is referred to by election officials as your "voting residence address." It is this address that defines your state and jurisdiction for voting. A P.O. Box cannot be used because it is not an actual residence address. You can only have one residence/domicile at any given time.
Even if you never voted from that address when you lived there, that is the address that you must use.
You don't need to have any current ties with your previous address or state. There is absolutely no requirement for overseas voters to continue to maintain a residence or to own property in the U.S. in order to vote.
In addition, it doesn't matter where you currently receive mail in the U.S. or how long you've been away.
For example, if you used to live in an apartment in Boston 30 years ago before moving to France to live, you would use your old apartment address in Boston for voting purposes. It doesn't matter if your parents have been receiving your mail at a different address or if you have no connection to Boston anymore.
Each person has one and only one residence, or domicile. When a civilian moves from one place to another, with the intent to make the new place home for a significant time, the individual's domicile changes to the new place.
If you leave place A to go to place B for a temporary purpose, with the intent to return to place A or to move to still another place, that does not change your residence/domicile. For example, Jane lived and worked in Ohio, but she visited family for one month in Maine before moving overseas. She last established residence/domicile in Ohio because she was only visiting Maine and didn't make her home there. Therefore, her voting residence address as an overseas voter will be in Ohio.
If you intend to live outside the United States for the long term and you aren't maintaining your domicile in your previous state of residence, under federal law you maintain the right to vote for federal offices (President, Vice President, U.S. Senator, and U.S. Representative). That federal law is called The Uniformed and Overseas Citizens Absentee Voting Act (UOCAVA).
To be eligible to vote for state and local offices (for offices such as Governor, State Legislator, County Clerk, Mayor, etc.), typically the voter must maintain residence/domicile in his/her last state of residence even when living overseas. For more details, you will need to check the laws of your state.
When you register to vote and/or request an absentee ballot using the Voter Registration/Absentee Ballot Request form, your local election official in the U.S. will send your ballot and voting materials to your overseas address, not your former U.S. address.