Couple of things.
Passover proper, the 14th of Nisan, is never defined in Scripture as a Sabbath, although it is a feast day. The first day of the Feast of Unleavened Bread is, however, referred to as a Sabbath type, a high holy day.
In the fourteenth day of the first month at even is the LORD'S passover. And on the fifteenth day of the same month is the feast of unleavened bread unto the LORD: seven days ye must eat unleavened bread. In the first day ye shall have an holy convocation: ye shall do no servile work therein. Lev 23:5-7
Holy convocation and servile work are the references to Sabbath.
In addition, John defines the High Day as a Sabbath (John 7:37, 19:31).
There is no necessary relationship of 14 days after the Spring Equinox. They measured it as the first full moon after the Spring Equinox.
So, to answer the OP, yes, there is always, unless the two days overlap, two Sabbaths of a weekly Sabbath and the first day of Unleavened Bread Sabbath in the week of Unleavened Bread.
For the first couple centuries, Alexandria taught that Passover the 14th was a Friday and the 15th was an overlap Sabbath of the Saturday and first day of Unleavened Bread. The church at Rome, however, taught the 14th was Thursday, so that Friday and Saturday were consecutive Sabbaths.
Back to John 19:31.
The Jews therefore, because it was the preparation [Friday the 15th for the weekly Sabbath], that the bodies should not remain upon the cross on the sabbath day [the first day of Unleavened Bread the 15th], (for that sabbath day was an high day,) besought Pilate that their legs might be broken, and that they might be taken away.
The key is John telling us it was also preparation day, which only happens prior to either a weekly Sabbath or Feast day Sabbath.
Does that answer your question?
EDIT TO ADD:
About the link to "refutation", the person doesn't understand, thus it's not a refutation.
On the Luke 6:1 reference, it is how they counted from the day after the weekly Sabbath to Pentecost. So, Luke 6:1 is referring to the end of the 2nd week or 14 days within the 50. It has nothing to do with a double Sabbath during Unleavened Bread week, but only as a means of marking the Sunday to the Sabbath as week one. Again, they were in week two.