Most people don't know this, but it is "rule airspace" above Class C up to 10,000'. That is, though you are not in Class C (and don't need to talk to them) you are required to have a transponder/ADS-B. SkyPuppy said he wanted to avoid the situation of needing ADS-B. So, overflying Corpus Christi (CC) is out unless he flies over 10K'. No, scratch that, you're in rule airspace if over 10K'. So, no overfly.
And I think his comment about staying within sight of land precludes him from doing a straight shot over Waldron to avoid Corpus Christi. (Which would also fly him thru Alert Area 632B.)
That leaves the next shortest route as flying between CC and Waldron, but having to stay at 1200' or lower, which would keep you under CC's outer shelf. I would follow the (what looks like a) land bridge over towards CC and then make a hard left to stay outside its inner circle. You don't have to stay over/near that land bridge but I wouldn't fly at 1200' over water for any distance over 1/8 mile myself. Fly south from there direct to your destination. You do not need to talk to CC on the radio, and you don't need a transponder/ADS-B. This also threads the needle between the Alert Areas.
Note that at 1200' and lower you would be in Class G airspace in this area -- so "clear of clouds" and 1 SM visibility would be your VFR requirements. But also note that while in G (uncontrolled airspace), ATC will supposedly not provide Flight Following, if so desired. Before or after this period you would presumably be higher (in Class E), which has different VFR cloud/visibility requirements, and is controlled airspace. I bring this up (besides the ATC consideration) because if 1200' was in Class E (which it is in most of the areas I fly in) then you'd need to stay 500' below clouds to be legal VFR -- which could potentially pinch you even closer to the water on a low-ceiling day.
The only other alternative I can see is going just west of CC and then south thru Alert Areas 632C & 632A.. The hours of operation are shown in iFly when you click on those areas. If you try to fly thru during those hours then as Jim pointed out, you need to keep your head on a swivel. Otherwise, and better, you should contact the controlling authority (NAS CC) and find out if these areas are going to be "hot" during your transit time. Or, probably easier, just use Flight Following (since ATC should know if they're hot).