Gulf Shores State Park at 496 campsites is a mega park. At a fantastic location.
They have water, elec (50A, 30A & 20A connections) and sewer for $475, $500, $600 per month depending upon location from Nov-Mar. There is a laundry facility near the store.
If you are an Escapees member - Rainbow Plantation is nearby at $325 per month + electricity costs.
Almost all private parks are going to meter and charge for electricity.
Just looking at RV Park Reviews - I count nine RV parks along the Gulf Shores beach section, 13 more south of US-98, 10 more between I-10 and US-98.
There are two across the bay on Dauphin Island, and six more south of Mobile/I-10.
I don't know if Meaher State Park in Mobile Bay has weekly/ monthly rates.