I know there are small furnished homes in Rockport for rent. A couple we know rented one for several months while they had their live-aboard sailboat in the yard for a major refitting. It was one block from the bay on S. Magnolia Street, not really a swimming beach there. It might be worth contacting a realtor to find out what is available, there are probably quite a few.
Places like Hunt's Castle offer winter rates as well as weekly and monthly rates, its right on the bay and has a fishing dock.
Hunt's Castle Hotel in Rockport, Texas - Rates
Personally, Rockport/Fulton has more of a feeling of a small ocean side community to me, along will all the amenities of a real town, whereas Port A is more of a tourist attraction with little feel of a community. Listing of cottages and homes
http://www.rockport-fulton.org/
If you would rather find something on a gulf beach I'm sure there are similar deals available over at Port Aransas.
Listing of cottages and homes
http://www.portaransas.org/cottages.html
It seems odd that Corpus does not have more development on Padre Island close to the beach, similar to Port A. But I have found a few, but I know very little about them.
El Constante Condominiums
20 reviews - 4 star
14802 Windward Dr
Corpus Christi, TX 78418-8024
(361) 949-7088elconstante.com
Gulfstream Condominium
23 reviews - 4 star
14810 Windward Dr
Corpus Christi, TX 78418-8003
(361) 949-8061gulfstreamcondos.com
Island House Condominiums
12 reviews - 4 star
15340 Leeward Dr
Corpus Christi, TX 78418-8021
(361) 949-8166ccislandhouse.com
Comfort Suites North Padre Island Corpus Christi
33 reviews - more info »
15209 Windward Drive
Corpus Christi, TX 78418
(361) 949-1112comfortsuites.com
Bahia Mar Corpus Christi
39 reviews - 4 star
15201 Windward Dr
Corpus Christi, TX 78418
(361) 949-2400bahiamarsuites.com
Beach Club Condos
52 reviews - 3 star
14721 Whitecap Blvd
Corpus Christi, TX 78418-7712
(361) 949-7577
This chamber of commerce website might be helpful
http://www.corpuschristicvb.com/accommodations.cfm
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