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Jack Schlossberg -- John F. Kennedy 's only grandson -- is looking to carry on his famous family's political legacy ... announcing a bid for Congress.

The 32-year-old Democrat posted a video on social media Tuesday night revealing he's running for New York's 12th congressional district -- the seat long held by Rep. Jerry Nadler , who's retiring after 34 years in office.

Check out the clip ... Schlossberg is seen buttoned up and speaks directly to the camera -- warning America is facing a "crisis at every level," and saying, "we can do better" by having the Democrats take back control of the House.

He hangs his hat on being a born-and-raised New Yorker ... and while he admits he might not have all the answers, he says he's ready to champion the voices of Manhattanites in W

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