I suggest that the week of Jesus' death did not involve a double Sabbath (as in two different days that were Sabbaths in the same week), but did involve two "Sabbaths" on the same day.
--
Festivals as Sabbaths
I agree with SLM that it's 15 Nisan, not 14 Nisan, that is a "Festival Sabbath". But in what sense are these Festival days Sabbaths?
Blumell & Wayment offer a helpful summary of the relevant Biblical texts:
Whenever the term Sabbath is used in John, it always refers to the actual
day of Sabbath (Friday sundown to Saturday sundown) and not to the beginning
of a festival held on another day of the week: John 5:9–10, 16, 18, 7:22–23, 9:14, 16.
However, in the Old Testament, “Sabbath” is occasionally used as a reference for
some festivals: Feast of Trumpets, Feast of Tabernacles (see Leviticus 23). Passover
is described as a day on which no work/labor should be performed (Sabbath-like).
This does not mean, however, that Passover was necessarily referred to as a “Sabbath” regardless of the day of the week it occurred, such as a Thursday or a Friday (see here pp. 78-79)
This is why some will take issue with calling 15 Nisan a Sabbath...it is and it isn't, depending on how we scope "Sabbath".
--
High Day
John tells us that this particular Sabbath was something special, which he called a "high day":
The Jews therefore, because it was the preparation, that the bodies should not remain upon the cross on the sabbath day, (for that sabbath day was an high day,) besought Pilate that their legs might be broken, and that they might be taken away (John 19:31)
I would be interested in any competing examples, but to my knowledge "high day" is never used, in any ancient Jewish or Christian source, to refer to a Sabbath that is not both:
- The weekly Sabbath (i.e. Saturday) AND
- A holy festival day
Thus, I understand John's meaning to be that the next day wasn't just "Saturday", it was both "Saturday" and 15 Nisan, the day the Passover lamb was to be eaten. It was therefore a particularly holy day.
--
The day of preparation
All 4 Gospels indicate that Jesus died on "the day of preparation": παρασκευή. παρασκευή does not refer to preparing for just anything; when παρασκευή is used to refer to a day, it always refers to Friday (see further discussion here).
--
Conclusion re double Sabbath
If we grant the broad notion of "Sabbath" that includes Festival dates like 15 Nisan, this means that the week of Jesus' death did not have two Sabbaths on different days, but it did have two Sabbaths on the same day, Saturday 15 Nisan.
Appendix--the date of Jesus' death
The OP noted that solving this problem would be very helpful in determining the date of the crucifixion--I agree, it is. If we conclude that the day following the crucifixion was a Saturday, 15 Nisan, then the crucifixion occurred on Friday, 14 Nisan.
If the Jewish calendar was correctly implemented (they were using lunar observation rather than calculation at this time, so they could have gotten it wrong--but to their credit, ancient astronomers were quite good at lunar observation, they did a lot more of it than we do, and Judea is not known for frequent overcast days), then we need only look for a 14 Nisan that would have fallen on a Friday during the tenure of Pontius Pilate (in office AD 26-36).
Humphreys & Waddington provide a helpful review of the lunar calculations & rules here, and in so doing, demonstrate that only 14 Nisan AD 30 or 14 Nisan AD 33 are possible.
AD 30 & AD 33 are indeed the most popular years among historians for dating the crucifixion. For my own assessment of the balance of evidence between these two possibilities, see my video series Chronology in the Life of Jesus.
As argued in the post linked by the OP, then, the Last Supper was not eaten on 15 Nisan, but Jesus really did die just before sundown on 14 Nisan--at the same time the Passover lambs were being slaughtered. John is careful to point out that Jesus is quite literally The Lamb of God.