You can either go up the ditch, from Corpus to Galveston there's plenty of depth - it ranges from 10 feet to deeper. Off shore is fine as well - it's a broad reach with the prevailing S or SE winds. This coming weekend we are going to sail Paloma from Port Aransas to Freeport, a short hop of about 150 miles. For you, it's another 40 miles from there to the Galveston Light, then 6 miles up the Galveston jetties and then the long trip up Galveston Bay.
Watergate is okay, but has a lot of tourists walking the docks and things disappear - it's also a bit of a motor through the channels to get back to the bay for sailing. You can't sail in Clear lake and you must stay in the channels until you get into the bay or you will run hard aground. You might check Blue Dolphin or Seabrook Ship Yard for a slip, they are both more private than Watergate and much closer to the bay.
s/v Paloma, Bristol 29.9, #141
Slipped in Bahia Marina, easy access to Corpus Christi Bay and the Gulf of Mexico.