Should We Visit Galveston & Corpus Christie?

Depends. Where are you coming from? Galveston has history and is close to NASA. Corpus has much better beaches and is a stop on the way to the Rio Grande Valley.

Corpus has the USS Lexington and a wonderful aquarium plus a nice water front.

Corpus has more, especially if you include Port Aransas and Padre Island where you can drive on the beach and cross over on a TDOT ferry.

Galveston also has the USS Cavalla museum (WW II area submarine) and the historic Strand shopping area.

Moved from Class C.

Across the bay from Galveston is the Bolivar peninsula where you can drive that class "C" right onto the beach and camp. Cost $10.00 a YEAR. Great place. Free ferry ride is nice as well
link to area info is http://crystalbeach.com/
BY the southern route through Louisiana did you mean I10 or the Blue Water Highway along the coast?

I like Rockport on the Gulf and Goose Island State Park as a place to stay.

First--the Gulf Coast and the Texas Coast are not the Washington Coast line with mountains, etc. But having seen both and only live 150 miles from Corpus, I like the Texas coast in the fall. Birding in Rockport is very good--get a map from the visitors center on where to go, great gulf seafood--different from Washington seafood, Port Aransas is a nice small coastal town with several good RV Parks, and Corpus has a bigger city feel. Enjoy the time there.

I've been to Corpus Christi twice. The first time was aboard the USS Lexington aircraft carrier, although we never tied up. We weren't allowed to. The Commanding Officers got into some sort of hissy fit.
The second time, I was a civilian and the Lexington was then a museum. I remember thinking at the time that Texas doesn't raise cattle - they raise oil wells.

Galveston has.

Historic Strand area shops and Restaurants.
Alissa old sailing ship you can tour.
Pleasure Pier (Amusement Park) On pier extending into the Gulf.
Water Park (Schlitterbahn)
WWII Submarine and destroyer escort on Pelican island.
Moody Gardens (Tropical Rain Forest)
Aircraft Museum next to Moody Gardens.

Take free ferry across to Bolivar and you have.

$10 beach camping.
Roll Over Pass some of the best bank saltwater fishing in the country.
Unique shops to browse.
Very good seafood restaurant (Stingarees) right on the inter coastal canal watch the traffic on the canal while you eat.

Corpus has

Much Better Beaches
WWII Aircraft Carrier (Lexington) now a museum.
Texas state Aquarium (excellent) has Dolphin show that is great you are close so be prepared to get wet!
Padre Island National Seashore.
A drive down Shoreline drive is worth doing.
Port Aransas is also close and offers some great fishing opportunities.