Several US states touch the Gulf:
Texas, Louisiana, Mississippi, Alabama, and Florida.
There is no such thing as a"Gulf of New Mexico". There is a "Gulf of Mexico", and Texas is the state that borders both the gulf and the country of Mexico.
It is the name of a peninsula in southeastern Mexico which touches both the Gulf of Mexico and the Caribbean Sea; it is also the name of a federal state which capital is the city of Merida. See related questions for further details.
Only Florida borders both the Gulf of Mexico and the North Atlantic Ocean.
The state of Florida
There is no US state that touches both oceans. The only country that has shores on both oceans is Chile in South America. (Mexico and 5 Central American countries border the Pacific and the Gulf of Mexico or Caribbean Sea, arms of the Atlantic.)
Florida is the only state to touch both the Atlantic and the Gulf.
mississippi and alabama both border the gulf
Post Oak Belt, Gulf Coast Plain, and South Texas Plains
The Pacific Ocean qualifies as such. All of Mexico's western shores are touched by such ocean. Indirectly, the Atlantic qualifies as such too, if you consider both the Gulf of Mexico and the Caribbean Sea as offshoots of such ocean.
West coast of the United States.
Florida is a state that is close to both the Gulf of Mexico and the Atlantic Ocean. It has a long coastline that borders both bodies of water, making it a popular destination for beachgoers and water enthusiasts.
The United States touches the Pacific Ocean, Atlantic Ocean, Gulf of Mexico, and the Sea of Cortez. Sea and ocean are both interchangeable terms, so each body of water that touches the U.S. is listed.