Which is more preferential: fasting the six days of Shawwal, or completing the five remaining days of a vow, given that it was impossible to complete them in Rajab and Sha'ban and Ramadan has now begun?

You must first complete the vowed fast, as it is obligatory, then fast the six days of Shawwal, which is supererogatory. If you intended the vowed fast to be continuous, then you must complete it consecutively; otherwise, you are only required to fast the remaining days of it. Vows are generally discouraged, as they burden the soul without altering destiny. However, once a Muslim makes a vow to perform an act of obedience, he or she is obligated to fulfill it.