Actually, you are doing it pretty smart IMHO. Low tides are usually around 6-8 pm, and that exposes another 50-100' of wet sand, the first cut and sometimes the first bar. If the 350 is waterproof then hit the shore side of the first bar, the gully or cut and dig everything. A long handled sand scoop is a must. On the dry sand work the umbrella lines. Usually after a heavy traffic weekend they have 2, sometimes 3 lines of umbrellas. I usually find coins in both places but the water seems to be better on getting rings, ear rings, bracelets, chains, cell phones, etc.. After dark, it is REALLY dark there!
Stewart Beach is straight in front of where I45 hits the gulf. Galveston has some sort of BS parking permit for the seawall now, some sort 800 # cell phone app. I have no idea how they enforce it.
Good luck and I hope the trip is a great one.