Assalaamu alaykum. If I pray a three-unit-of-prayer Witr prayer with one sitting, should I do Tawarruk (sitting with the buttocks on the ground whith both feet on the right side of the body) in that last sitting or not?

All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger. 

The scholars held different views regarding the Tawarruk in prayer. The Shaafi'is held the view that Tawarruk should be done in every Tashahhud after which there is Salaam. Accordingly, you should do Tawarruk in the Tashahhud that you mentioned.

However, according to the Hanbalis, Tawarruk is only prescribed in the prayer that has two Tashahhuds, so (according to this view) you should not do Tawarruk in the Tashahhud that you mentioned.

It appears that the view of the Hanbalis is the preponderant one, Allah willing.

Shaykh Ibn ‘Uthaymeen was asked in Liqaa’ Al-Baab Al-Maftooh, "When should a worshipper sit in the position of Tawarruk in prayer, and in which prayers?"

He replied, "Tawarruk is done in the final Tashahhud of every prayer that has two Tashahhuds; i.e. the last Tashahhud in the Maghrib, the ‘Ishaa’, the ‘Asr, and the Thuhr. As for the two-unit prayers, such as the Fajr and the regular Sunnah prayers, then there is no Tawarruk in these prayers. So Tawarruk is only in the last Tashahhud of every prayer that has two Tashahhuds." 

As for the Hanafi scholars, the Tawarruk is not prescribed in principle [according to them]; and according to the Maaliki scholars, it is prescribed to do Tawarruk in every sitting in the prayer, but this manner of performing the Witr prayer is not prescribed in their school of Fiqh (as the only prescribed way of performing Witr, according to them, is to pray two Rak‘ahs and recite Tasleem and then perform a third Rak‘ah on its own with an independent opening Takbeer).

Allah knows best.