Having checked the genre on maqtal (i.e. books on the tragic events of Karbala) we could not establish that Imam al-Husayn (a) – may our lives and that of our children be sacrificed serving his mission, was beheaded in the state of sajdah. The nearest we could get is what is found in the book The Story of Karbala by Ali Nazari Munfarid (forward by Ayatullah Nasir Makarim Shirazi, tr. Sayyid Hussein ‘Alamdar) under the title. “The Last Moments of Imam (AS)”,
‘And when Imam (AS) fell down from the horse back upon the earth, he wanted to lie down upon his right side, but because of too many injuries it wasn’t possible. Then he tried to lie down on his left side but couldn’t do so; thus he collected some sand and dirt around him, made something like a pillow and placed his head upon it. The Kufan army was bewildered as to what state is he in? Some said: “He is already dead, “ while others said: “He doesn’t have the strength for fighting.” (p. 302).
Only Allah knows, if the Imam (a) here was performing a sajdah or simply resting his head on the pile of sand.
As for your question whether the Zuhrayn were prayed separately, it seem so. Most maqtal mention about the Imam (a) and his companions offering their zuhr prayer on ‘Ashura day but do not say anything about ‘asr prayer. Also, when the Imam (a) and his caravan were with Hurr b. Yazid Riyahi and his troops at Dhu Husm on 27th Dhil Hijjah, 60 AH, the Imam (a) led zuhr and ‘asr prayer separately. These prayers were joined by Imam’s caravan as well as by Hurr and his army.
Allah knows the best.