It seems to be "酔う" in the second sense:
〔船などに〕feel sick
船(車,飛行機)に酔う get 「seasick (carsick / airsick)
彼は船に酔いやすい(酔わない) He is a 「poor (good) sailor.
The fact that little by little everyone is getting further off the tempo seems to be what makes him nauseous. He's also complaining that no-one is watching him, but I don't see why that would make him feel sick.
Here's the second example sentence above:
"ほら、海って波の音がするじゃないですか。私その波の音に酔っちゃうんですよ。あのザザーッて波の音がダメですね。酔ってゲロ吐きそうになっちゃいますね。"
"You know, [when you are near] the sea, the waves make a sound, right? The sound of the waves makes me [sea] sick! I really can't handle that rushing sound of the waves. It makes me nauseous, and I feel like I'm going to really throw up, you know?"