What is the jurisprudential stance and what is required to be done, and is expiation obligatory for one who swore an oath of prohibition upon herself for someone other than her husband using the phrase: "My back and my belly are forbidden to anyone other than him," and would her cohabitation be unlawful if she were to marry another person?

You must repent to Allah for the unlawful acts you have committed by abandoning the sin, regretting it, and resolving not to return to it, while maintaining discretion and not publicizing it. The statements you made do not prohibit marriage to this man or anyone else. You owe an expiation for an oath due to the differing opinions of scholars, and the safest is to perform it by feeding ten poor persons or clothing them, or by fasting three days if you cannot find the means.