Adoration is a profoundly personal experience so there is no “correct” way to pray, beyond of course keeping reverent before the lord and quiet (common decency to others).
Generally what I would recommend in praying during adoration is firstly giving a mental act of contrition. Reflecting in an attitude of repentance on ones own detestfully sinful nature before the perfection of God and on his great and unfathomable mercy in his sacrifice for us. Following this I would reflect on the mystery and glory of the gift of the eucharist and on the real presence of Christ before you, on the incomprehensible majesty of God with us, to orientate yourself into a proper attitude before the lord.
Following this the prayers would be your choice, but personally I would pray the sanctus repeatedly as a very appropriate prayer before God interspersing this timeless prayer with reflection in an open mind to listen to God’s voice in mental stillness. (simply constant reptition could lead to a lack of meaning and you lose the important aspect of reflection and contemplation) You could also engage in a “mental dialogue” in these interspersions reflecting on God in your life and engaging in a conversation, Often reflecting on oneself and on God leads to an awareness of things that we miss in everyday life. (many saints have had profound revelations in eucharistic adoration)
As to the rosary, I personally would not pray the rosary since I see adoration as focussed solely on Christ and on contemplation. To me personally although the rosary is good, it is still missing the point in my opinion of adoration. However saying that, it is my personal opinion and the rosary is definitely permitted and perfectly fine should you wish to do so, it is simply not for me the best way to spend adoration.