The one who is doing Qiraan, i.e., combining Hajj and ‘Umrah, only has to do one tawaaf and one saa‘i, like the one who is doing Ifraad (Hajj only).
The tawaaf that is obligatory in his case is tawaaf al-ifaadah. With regard to tawaaf al-qudoom, it is Sunnah. In his case, saa‘i may be done after tawaaf al-qudoom, or he may delay it and do it after tawaaf al-ifaadah.
This is what is indicated by the saheeh Sunnah and is the view of most of the scholars.
Ibn Qudaamah (may Allah have mercy on him) said: The well known view narrated from Ahmad is that the one who is doing qiraan, i.e., combining Hajj and ‘Umrah, does not have to any actions but those that are required of the one who is doing ifraad (Hajj only), and one tawaaf and one saa‘i are acceptable in his case, for both his Hajj and his ‘Umrah. This is the view of Ibn ‘Umar and Jaabir ibn ‘Abdullah, and was also suggested by ‘Ata’, Tawoos, Mujaahid, Maalik, ash-Shaafa‘i, Ishaaq, Abu Thawr and Ibn al-Mundhir.
Another report was narrated from Ahmad, which suggests that he has to do two tawaafs and two saa‘is. This was the view of ath-Thawri and Abu Haneefah, and was also narrated from ‘Ali, but the report from him is not saheeh.
Then he quoted the evidence of the majority and said: It was narrated from ‘Aa’ishah (may Allah be pleased with her) that she said: As for those who combined Hajj and ‘Umrah, rather they only did one tawaaf for both. Agreed upon. In Muslim it is narrated that the Prophet (blessings and peace of Allah be upon him) said to ‘Aa’ishah, when she did Hajj and ‘Umrah together (qiraan): Your tawaaf is enough for your Hajj and ‘Umrah.”
It was narrated that Ibn ‘Umar said: The Messenger of Allah (blessings and peace of Allah be upon him) said: “Whoever enters ihram for Hajj and ‘Umrah (combined), one tawaaf and one saa‘i is sufficient for both.” It was narrated from Jaabir that the Prophet (blessing and peace of Allah be upon him) combined Hajj and ‘Umrah (qiraan) and did one tawaaf for both. These two reports were narrated by at-Tirmidhi, who said concerning each of them: a hasan hadeeth. Ibn Maajah narrated from Jaabir, Ibn ‘Umar and Ibn ‘Abbaas that the Messenger of Allah (blessing and peace of Allah be upon him) and his Companions only did one tawaaf for their ‘Umrah and Hajj. End quote from al-Mughni, 3/241
Shaykh Muhammad al-Ameen ash-Shanqeeti (may Allah have mercy on him) said: The majority of scholars who differentiate between qiraan and tamattu‘ say that for the one who is doing qiraan, one tawaaf is sufficient for both his Hajj and ‘Umrah, and that is tawaaf al-ifaadah, and one saa‘i. They quoted as evidence saheeh hadeeths, which cannot be matched by those who differ from them.
End quote from Adwa’ al-Bayaan, 4/430
Thus it becomes clear that what you were told about the one who is doing qiraan of having to do one saa‘i is the correct view.
It is quite clear that when the one who is doing qiraan has done tawaaf upon arrival (tawaaf al-qudoom) and saa‘i, he then remains in his ihram and does not remove any of his hair and does not call what he has done ‘Umrah; rather it is part of the actions of Hajj and ‘Umrah combined.
And Allah knows best.