Is it possible for a U.S. State to become a territory?

Possible but unlikely. It is very difficult for a state to dissolve the bonds of statehood (even though a few Texans claim otherwise). I believe that the request of a state plus an act of congress could give a state legal permission to leave.

A note on Texas from http://forttumbleweed.net/statehood.html

"When all attempts to arrive at a formal annexation treaty failed, the United States Congress passed, after much debate and only a simple majority, a Joint Resolution for Annexing Texas to the United States."

"Under the terms, Texas would keep both its public lands and its public debt. Texas would also have the power to divide into four additional states 'of convenient size' in the future if it so desired, and it would deliver all military, postal, and customs facilities and authority to the United States government. Neither this joint resolution or the ordinance passed by the Republic of Texas' Annexation Convention gave Texas the right to secede."
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I don't think so. A serious attempt would likely lead to another civil war. You should consider moving to the United States Virgin Islands if you would like to live in a territory instead of a state.