Which days are we to have a “Shabbatown”? Which days are we to have not only a “Shabbatown”, but also “Shabbat Shabbatown”?

Answer From The Annual Reading Schedule - Week 28 Notes For Leviticus:

The phrase "Shabbat Shabbatown" ("Sabbath rest" or "Sabbath of rest") occurs in the following Scriptures:

Exodus 31:15 – In reference to the weekly 7th Day Sabbath
Exodus 35:2 – In reference to the weekly 7th Day Sabbath
Leviticus 16:31 – In reference to the annual Day of Atonements (Yom HaKippuriym)
Leviticus 23:3 – In reference to the weekly 7th Day Sabbath
Leviticus 23:32 – In reference to the annual Day of Atonements (Yom HaKippuriym)
Leviticus 25:4 – In reference to the 7th year land rest once every 7 years

The word "Shabbatown" ("rest") by itself occurs in the following Scriptures

Leviticus 23:24 – In reference to the annual Day of Blowing of Trumpets / Remembrance of Blowing of Trumpets (Yom Teruah)
Leviticus 23:39 – In reference to the 1st Day of the Feast of Tabernacles/Booths (Sukkot) and the 8th Last Great Day
Leviticus 25:5 – In reference to the 7th year land rest once every 7 years

The phrase "Shabbatown Shabbat-Qodesh" (which means, "a rest day, a ceasing-Set-Apart" or "a rest, a Sabbath Set-Apart" occurs in the following Scriptures:

Exodus 16:23 – In reference to the weekly 7th Day Sabbath

In summary, the days and times when we are told to have a "Shabbatown" (a rest) in the Hebrew, but are not called "Shabbat Shabbatown" are:

Yom Teruah (1st day of 7th Biblical Month) – Leviticus 23:24
1st Day of Sukkot (15th Day of 7th Biblical Month) – Leviticus 23:39
8th Last Great Day (22nd Day of 7th Biblical Month) – Leviticus 23:39

In summary, the days and times specifically referred to as times for having a "Shabbat Shabbatown" (Sabbath of rest) in the Hebrew are:

Weekly 7th Day Sabbath – Exodus 31:15, Exodus 35:2, Leviticus 23:3
Annual Day of Atonements - Leviticus 16:31, Leviticus 23:32
7th Year Land Rest – Leviticus 25:4-5