Actually, I believe they approved the Declaration on July 2 and didn’t sign it till early August.
Here’s what John Adams wrote to Abigail:
“The second day of July, 1776, will be celebrated by succeeding generations as the great anniversary festival. It ought to be solemnized with pomp and parade, with shows, games, sports, guns, bells, bonfires, and illumination, from one end of this continent to the other, from this time forward forevermore.”
But I’m sure he would be fine with the 4th.
The original independence resolution was passed on 7/1 with nine in favor, two against, one delegation split, and one abstention.
On 7/2 the vote was taken again. This time it was "unanimous" (with one abstention).
The Declaration was "adopted" on the 4th. I've never been completely sure what that means. I don't know if they took a vote on it or just finished critiquing it so that there were no further objections. Only John Hancock and Charles Thomson signed that day.
On 8/2 fifty members of the Congress signed the freshly drafted parchment with the final wording. This is the "signing" most people think of.