Why did she say "Lo Adoni"?
Brachos 31b, cited in Rashi: You (Eli) are not a master in this matter (of my prayer), nor is Ru'ach ha'Kodesh upon you, for you judged me unfavorably 1 . (I am not drunk!)
Aderes Eliyahu: Old texts of Rashi say 'Lo Adoni k'Sarah.' The printers did not understand this, and omitted it.. Midrash Shmuel Rabasi says 'k'Sarah Rivkah Rachel.' Eli was astounded to see her lips moving, with no sound exuding. He knew that she was a Tzadekes, a Navi's wife! He asked the Urim v'Tumim 2 ; the letters Hei, Chaf,, Shin and Reish lit up. With Ru'ach ha'Kodesh, the Kohen knows how to arrange the letters (Ramban Shemos 28:30). Eli arranged them to spell Shikorah. Chanah said, you lack Ru'ach ha'Kodesh; they spell k'Sarah! I am an embittered woman praying for children, like her.
What are "Yayin v'Shechar"?
Radak citing Targum Yonasan: They are new and old wine.
Radak #2: Wine, and [intoxicants made from other] Peros.
What is the meaning of "va'Eshpoch Es Nafshi Lifnei Hashem"?
Radak: Through excessive prayer, it is as if she poured her soul. This is like "Shifchu Lefanav Levavchem" (Tehilim 62:9).
Radak citing Targum Yonasan: I said that I will humiliate my soul in prayer in front of Hashem.
Malbim: [I am not drunk. I prayed quietly because] I prayed to Hashem. (I prayed at length due to my pain.)