There is a park off of Austin Hwy and Corrine St.
Circle B Rv Park. It is a mostly full time park of workers. Not luxurious, but well-paved streets and conveniently located off of I-35 and Loop 410. Very reasonable and if you are staying for a week, they will negociate the price. Their electric is 100%. They also use 150 solar panels to run their park which I like apart from being grid tied.
From the park, you take Austin Hwy to Broadway and right to the front of the Alamo. You can find the park on Google Earth with the address.
I have a rental nearby and use this park when I stay. It is the only park located in the central part of San Antonio. Trust me, there is much more than the KOA. It is a safe park and we use it regularly when we visit SAT. It can't be beat. Worth checking out.