Did carole king and James Taylor have an affair?

No.

She's currently married to a man named Rick Sorenson. She's also been married 3 previous times:

Gerry Goffin 1960-1968, 2 children, marriage ended in divorce

Charles Larkey 1970-1976, 2 children, marriage ended in divorce

Rick Evers 1977-1978, no children, marriage ended as a result of Ever's death

Rick Sorenson 1982-present, no children
No, Carole King and James Taylor were never married.

She's currently married to a man named Rick Sorenson. She's also been married 3 previous times:

Gerry Goffin 1960-1968, 2 children, marriage ended in divorce

Charles Larkey 1970-1976, 2 children, marriage ended in divorce

Rick Evers 1977-1978, no children, marriage ended as a result of Ever's death

Rick Sorenson 1982-present, no children