Lady Bird Johnson City Park is where most prefer to stay. It is located near the airport (little air traffic), at the golf course (may be closed due to reconstruction) and has a bit more of a park-like setting with lots of trees and grass. It is a little further out (2 miles from downtown) but not as far out as the KOA.
Closest to downtown is the Fredericksburg RV park which is a well run parking lot with nice meeting rooms and such.
Oakkwood is about a half mile closer than Lady Bird Johnson and has a small, rather tight but doable, area in the trees then the balance is just a parking lot. The pool there is large if it is open.
Hill Country RV Park might be a lower price option but is strictly a parking lot and is about the same distance from downtown as Oakwood.
We have stayed in all of these parks and survived well.
The KOA is a few miles out of town and we have never stayed there. Michelle has been in the park and tells me it is nice. The Stonewall park is several miles from town. It is close to the LBJ Ranch which is interesting. There is nothing in Stonewall but it is central to several things in the area including Luckinbach. There is also a National Park Service exhibit in Johnson City on the LBJ presidency which is quite informative. We have stayed at the Stonewall park and it is a nice, well run parking lot.
Have fun!