This is where AAA and Good Sam differ.
AAA will find a truck to tow you. It is up to you to decide where to be towed.
Good Sam will first attempt to locate a shop that will accept your vehicle. IF and WHEN they find that shop, they will then locate a tow truck to tow you to the shop. If they cannot find a shop willing to accept your vehicle, they will not send out a tow truck.
With both, if you need other than a small size wrecker (standard auto tow truck), it can take longer to get that tow truck to you because there are fewer available.
As for what happens now, GS will most likely agree to pay your out of pocket tow costs and will offer an additional year of service for free - it's their standard "compensation package".