If you're in the US - then no. You'll owe payment on your purchase on your next statement due date (September 12th), not your current due date (August 12th). In other countries, your closing date and due date may be the same, in which case you're required to make a payment on the same day your statement is posted.
Your credit card has two different dates that you need to know:
Closing date - The date your statement for the current period is posted.
Due date - The date your payment is due.
After your statement is posted on the closing date, you usually have a grace period of at least 21 days (since the passage of the Credit CARD Act of 2009) before you're required to make at least the minimum payment on the statement's balance and before interest begins accruing on your balance. If you make a purchase after the closing date but before the due date, this purchase will be recorded on the statement for the next period.
In your case, your due date is August 12th. Your closing date is probably several weeks before that. Let's say it's July 20th. If you make a purchase anytime between July 20th and August 20th, this purchase will be recorded on your next statement, which will post on August 20th. Your payment for that purchase is due on September 12th, the due date for that period.
I'm assuming that you're paying in full each period (as you indicated in your question), because if you don't, then obviously, portions of your balances from previous statements will appear on your next statement(s) because you haven't paid them off in full yet. For some credit cards, the grace period only applies to the balance from the current period, so you might still accrue interest on balances from previous periods every day that you don't make a payment.
Note that your balance before you make a payment may include the balance of your last statement and the balance of your purchases after the statement is posted, but if you select the option to pay in full, the amount you pay will be the balance of your last statement, not the total current balance on your card. In general, it's the balance when your statement is posted that matters.
I'm looking to get this date moved to later on in the month, how do I do that?
This varies depending on your credit card provider, but it's probably somewhere in the online interface, or you could call their customer service and ask them to change the date for you. The change may not take place immediately, to prevent you from pushing back your payment for the current period, but it will probably take effect within a few cycles. Some credit card companies won't let you change this date during your card's anniversary period, so you may need to wait until the next period to change it.