Ok guys - After quite a bit of feedback and more investigation, it looks like we are gonna go back to Galveston State Park this time. (See the comments section if you`re curious about this decision.)
Admission Fee: $5.00 per adult – children under 12 are free. Be sure you stop at the office and pay if the booth is closed. They can ticket your vehicle if your paid fee receipt isn’t attached to your windshield.
After you enter the park and pay your entrance fees, park in the lot located to the left of the park entrance. Cross the dunes and let's meet down on the beach.
We can walk down to the Jamaica beach area or stay within the area of the state park - basically wherever we have enough run to let the dogs play. The forecast is for low 80's on the island. Don't forget to bring some water for your pup and a long line if you don't want to let them off leash. (Keep the county wide leash rule in mind if you do decide to let your dog off leash.)
As discussed in the comments section, Galveston has an ordinance against dogs on restaurant patios, but I did find out about some available options for eating after we have fun at the beach. Several of them are listed in the comments section. If you know of any others, please let us know.