Do I need to go to the Immigration office immediately after failing to report my 90-day stay in Thailand?

It's your choice. The window of 90 days report in person at the local immigration office is 15 days before the due date until 7 days after the due date (calendar day). This is the period you can do 90 days of reporting without paying a fine.

Once you pass this 7-days-after period, the fine of 2000 thb stays the same anytime until you do 90 days report next time. In this case, you can do it as soon as possible, or wait until September, any way you pay the fine. Be ready to be grilled by the officer if you want to wait until the next extension application.