To the contrary, God's sovereignty does mean that He has sovereign dominion over everything. Sovereignty at its basic level of meaning is to be in control of something. So biblically God's sovereignty refers to his complete control of the universe.
The Westminster Confession of Faith gives an excellent summary of this understanding of God's sovereignty: 'God from all eternity, did, by the most wise and holy counsel of His own will, freely, and unchangeably ordain whatsoever comes to pass' (
Ch 3). To support this position, the WCF provides this Scripture:
EPH 1:11 In whom also we have obtained an inheritance, being predestinated according to the purpose of him who worketh all things after the counsel of his own will. ROM 11:33 O the depth of the riches both of the wisdom and knowledge of God! how unsearchable are his judgments, and his ways past finding out! HEB 6:17 Wherein God, willing more abundantly to shew unto the heirs of promise the immutability of his counsel, confirmed it by an oath. ROM 9:15 For he saith to Moses, I will have mercy on whom I will have mercy, and I will have compassion on whom I will have compassion. 18 Therefore hath he mercy on whom he will have mercy, and whom he will he hardeneth.
So God's sovereignty means much more than he holds the final judgment, as you say. Yes, sovereignty refers to governance but governance means 'to be in control of', 'to be in charge of'.
So God is in charge of, controls, what happens in our world.
Oz