Is it true that reciting specific surahs and verses from the Holy Quran, sometimes out of the order of the Mushaf, can expedite marriage?

There is no evidence for dedicating specific surahs from the Quran to facilitate marriage. Such a dedication would require Sharia-compliant proof. We advise you to continue supplicating, for every soul will complete its provision. There is no harm in seeking a husband through lawful and dignified means, or by seeking help from friends and relatives. You did well by refusing to meet the suitor outside the home, to prevent temptation.