You don't have to prove anything to the IRS when you file your tax return. You just report the sale and take the exclusion. You only report the dates of purchase and sale, and the cost and selling price. You don't report the dates of residence or the date you moved. You would only have to prove that you met the residence requirement if you got audited.
If your gain is $250,000 or less, you might be able to sign a form at the closing of the sale certifying that you qualify to exclude the entire gain. Then the closing attorney or agent won't issue a Form 1099-S, and you won't even have to report the sale on your tax return.
I don't know what kind of proof the IRS would require in the unlikely event that you got audited, but I think that having some evidence of when you actually moved would be a big help. How did you move your furniture? If you used a moving company, do you have a contract or receipt? If you rented a truck, do you have a record of the rental? If necessary, you could also ask neighbors of either house for a letter saying when you lived there or when you moved.
