Is the aforementioned sin—committing adultery after a formal engagement and the recitation of the Fatiha due to difficulties in marriage and fear of falling into widespread indecency with someone other than one's partner—considered among the major sins, and does the intention of seeking protection from falling into wider indecency mitigate its prohibition?

If by "official engagement" and "recitation of Al-Fatiha" you mean that a valid marriage contract has been concluded, then the woman is considered your wife, and whatever is permissible for a husband with his wife is permissible for you with her. The absence of a wedding procession (zaffah) or a delay in paying the dowry does not affect this. However, if what occurred was merely an engagement, then the woman is still a stranger to you. You must fear Allah and refrain from mixing with her. Remember that fornication with this woman is a grave indecency and a major sin, and that Satan has deceived you. Therefore, you must repent, feel remorse, and immediately cease this disobedience.