What is the correct course of action if a young man proposes to a girl who is several months older than him, and the girl fears that the young man's family will not accept her and her family due to the age difference, which could cause family problems?

The age difference does not affect the validity of the marriage contract. Therefore, do not hesitate to accept marriage from a righteous young man due to the close age. Inquire about his religion and character from trustworthy individuals, and if they praise him, then seek Allah's guidance (perform Istikhara). If his parents object, he should try to persuade them. The principle is the obligation to obey them unless a preponderant interest contradicts it. Entrust your matter to Allah Almighty and supplicate to Him to facilitate your marriage to a righteous man. If marriage to him is not destined for you, then seek someone else. It is permissible for a woman, according to Islamic law, to seek a righteous husband and offer herself to him within legitimate parameters.