Adonai is used as a description of God or as a title of respect, rather than a name in the proper sense. David would have had no problem addressing God as “Adonai” and then addressing Saul as “Adonai” as well—no one would have been confused by that, just as no one today thinks that referring to the English poet as “Lord Byron” ascribes deity to him.
Adonai is certainly an apt title for God. While there are other lords who hold sway over limited realms, God is the Lord of all the earth