Is swearing falsely by Allah—with the intention of escaping a specific situation—considered an oath that incurs damnation (yamin ghamus), and what is its expiation?

Your oath is a yameen ghamus, which is one of the major sins. The majority of scholars hold that there is no expiation for it except repentance, but it is more cautious to offer the expiation for an oath to avoid scholarly disagreement.

You must repent to Allah Almighty for swearing a false oath to please a created being. You must also avoid relationships between men and women online and repent for what has occurred through them, as they contain impermissible aspects.