Green’s Dictionary of Slang

(US) tipsy.

1914		R. Lardner You Know Me Al (1984) 33: They have beer out here that they call steam beer. I had a few glasses of it and it made me logey.
1920		Ade Hand-made Fables 97: Attend several going-away Dinners. As soon as you are feeling logy and oppressed, begin packing.
1951		(con. 1920) S. Longstreet Pedlocks (1971) 279: They sat stuffed and logy, looking at the adults.
2009		J. Stahl Pain Killers 51: I’d probably be too confused and logy to notice.