The American Ballet Theatre tonight announced it has "firm commitments" for financing and will begin its annual U.S. tour Dec. 6 at the Kennedy Center.

"The board just gave us the green light," said an ABT spokesman.

But ABT's struggles may continue for a while. Tonight, ABT sources said the board also decided to begin search procedures for a new artistic director to replace Mikhail Baryshnikov, who has confirmed plans to leave the company next spring for a movie role. Baryshnikov's contract expired earlier this year and has not been renegotiated.

On its financing, the ballet troupe said it now has commitments for all but $200,000 of an urgently needed $2 million--either as personal guarantees by board members for a bank loan or as promises from those who have pledged funds to the company to accelerate actual payment of their contributions.

In addition, the ABT board reaffirmed its plans to proceed with the $1.1 million full-length production of "Cinderella." The work will have its world premiere in Washington Dec. 20, where it will be presented for two weeks at the Kennedy Center.

In a resolution approved unanimously today, the trustees said they planned "to proceed with all aspects of its 1983-1984 season" and to apply to Manhattan's Chemical Bank for a loan, to be backed in part by personal guarantees from some wealthy ABT trustees.

The board's acting chairman, Melville Straus, however, warned in a prepared statement that "solving our short-term cash flow problems . . . in no way diminishes our ongoing need for substantial contributed support."