There's a "passport fair" at the Downtown Fort Worth Post Office on Lancaster on Sunday August 6th. These fairs are described as "passport acceptance facilities such as post offices, libraries, and local government offices are hosting special passport acceptance fairs across the United States to help you and your family apply for your passport." More about them at https://travel.state.gov/content/travel/en/News/passports/special-passport-acceptance-fairs.html.
This one doesn't require an appointment, but I'd still get there early. It doesn't make your passport free ($165 altogether) but does make obtaining one easier since there will be people to assist and walk you through it.
Hint. Try to park free at the Trinity Railway Express (TRE) parking lot. Technically you need to ride the railway to park there for free but if you act like you belong no one will notice. If that doesn't suit you, just park along the street near the station parking lot and use the tunnel to walk through the station and to the post office.