It is proven in the Sunnah that we are instructed to eat and drink with the right hand, and we are forbidden to do that with the left hand. It was narrated from Ibn ‘Umar (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “When one of you eats, let him eat with his right hand, and when he drinks, let him drink with his right hand, for the shaytaan eats with his left hand and drinks with his left hand.”
Narrated by Muslim, 2020
It was narrated from Jaabir ibn ‘Abdullah (may Allah be pleased with him) that the Messenger of Allah (blessings and peace of Allah be upon him) said: “Do not eat with the left hand, for the Shaytaan eats with the left hand.”
Narrated by Muslim, 2019.
If a person needs to use both hands, there is nothing wrong with that because of the report which says that the Prophet (blessings and peace of Allah be upon him) drank from a waterskin and a bucket, which usually needs both hands. Hence the scholars granted a concession allowing the use of both hands when necessary.
An-Nawawi (may Allah have mercy on him) said: Our companions said: if he drinks with both hands cupped and has a ring of silver on his finger, there is nothing wrong with that.
End quote from al-Majmoo‘, 1/316
Al-Qurtubi (may Allah have mercy on him) said in his commentary on the verse (interpretation of the meaning): “except him who takes (thereof) in the hollow [ghurfatan] of his hand” [al-Baqarah 2:249]: Some of the commentators said: al-gharfah means in the hollow of one hand and al-ghurfah means in the hollow of both hands. And some commentators said that both mean the same thing.
‘Ali (may Allah be pleased with him) said: Hands are the cleanest of vessels.
The one who wants to consume only that which is halaal nowadays with no doubts whatsoever should drink with his hands water from springs and rivers at any time of night or day.
End quote from Tafseer al-Qurtubi, 3/253, 254
If you carry the cup in both hands, you should mostly use the right hand with the support of the left, not the other way round, so that the left hand will not be the main one with the right hand following or supporting it.
And Allah knows best.