Although many of the "But God" statements of the Bible refer to the difference that God makes when He works, there are other times "But God" pertains to a response to God seen in one's heart and life. The following examples are given as space allows.

Nevertheless my brethren that went up with me made the heart of the people melt: but I (Caleb) wholly followed the Lord my God (Joshua 14:8).

Then said David to the Philistine, Thou comest to me with a sword, and with a spear, and with a shield: but I come to thee in the name of the Lord of hosts, the God of the armies of Israel (1 Samuel 17:45).

And David was greatly distressed; for the people spake of stoning him…but David encouraged himself in the Lord his God (1 Samuel 30:6).

But it is good for me to draw near to God (Psalm 73:28).

But seek ye first the kingdom of God, and His righteousness; and all these things shall be added unto you (Matthew 6:33).

See also Romans 4:20; 12:2; 1 Timothy 6:17; 1 Peter 3:15, etc.