Dulcibella is one of the later forms of the Middle English feminine name Douce, from the word dulcis, meaning "sweet" in Latin. The name went through many spelling options between the 13th and 16th centuries, including Douse, Dowse, Dulcia, Dowsabel and Dousabel. Later variants include Dulcea, Dulcina and Dulciana. Dulcinea was the form used by Cervantes for Don Quixote. Although Dulcibel is pronounced DULL-sih-bell or DOOL-sih-bell depending on your native language, the spoken form of the name for centuries was Dowsabel, pronounced DOW-suh-bell. Like some other medieval names, the form Douce was also used for boys, and the name has given us surnames like Dowson and Dowse.
At some point in the mid to late-1600's, Shakespeare picked up the name and turned it into a term meaning "sweetheart" in The Comedy of Errors, a little bit like how Doll is a nickname but also a similar term. The only other place I see Dulcibella pop up is in the children's novel The Riddle of the Sands, in which Dulcibella is a boat named after the author, Robert Erskine Childers' sister.
While the name is, and has always been uncommon, both in the U.S. and England, it has increased in popularity over time and still used sparingly in England, but has not been used in the U.S.