What is the ruling on the vow made by the questioner to herself not to marry anyone other than a specific person, and is expiation or fasting required after breaking this vow?

The questioner must repent and sever her illicit relationships with non-mahram (definition) men. If a suitor of good character and religious commitment proposes to her, she should accept him; otherwise, she should reject him. If the promise between them was not in the form of an oath, then there is nothing in it. However, if it was a promise to Allah Almighty in the form of "I promise Allah," then it appears to be an oath for which expiation is required if she does not fulfill it.