Because of this, I expected I might find it used where people were in particular reverence and awe of God. Sure enough, it's used at times when people are being called by God and have a particular revelation of who He is. Isaiah has a revelation of Adonai's holiness (Isaiah 6: 1 and 8) and responds with a deep awareness of his own sin before he answers willingly to God's call. Jeremiah likewise feels sort of overwhelmed (Jer 1: 6) and responds by telling the Sovereign Lord, Adonai, that he feels too young and inexperienced to speak on God's behalf.
So it's not surprising that it's also a name people use when acknowledging God's lordship and accepting His sovereign will, even when it's disappointing to them. Moses uses it (Deut 3: 24) when reluctantly accepting that he will not cross over to the promised land. David uses it (2 Sam 7: 22) when accepting that he will not be the one to build a temple for the Lord.
We can embrace God's sovereign will more easily when we are also convinced that He is good and caring. Adonai Tov (Good Lord) is another of His names that we find in the Bible.
Read on below for more about that.