Is expiation obligatory for someone who swore inadvertently – while prostrating – saying: “By Allah, I have repented to You that I will not disobey You, O my Lord,” and then committed a sin?

The قسم (oath) that you uttered unintentionally does not obligate you to an expiation (kaffarah), because it falls under the category of laghw al-yameen (idle or unintentional oaths), for which there is no expiation. Allah Almighty says: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended of oaths." You must beware of sins and repent from them, and whenever you sin, hasten to repent.