Well Charles, it sounds like you probably do have a problem in the steer gear box.
If you have a "605" box, a metric box that has the numbers '605' embossed in the casting, that is a very common problem... they are JUNK! I went through four of them in my dirt car before buying a good Saginaw based Sweet racing box. The 605 box could not take the punishment of dirt racing, and they would do the very same thing you describe only worse. They would overcorrect then lock to one side or the other. I had to get towed off the track on each one.
If you do have a 605 box in there, get it out and use it for a boat anchor!
You most likely have a 3/4" input shaft, as the 13/16" were mostly for trucks.
I am not sure what most unmolested factory GM factory steer box ratios are, but there is also another factor to figure in...
Be carefull on which valving you get on your next box!
A 185 valve is the softest and easiest to turn. Most road racers and dirt track racers want these if they are going to run a quick steer reducer, as it takes very little effort to move the wheel.
A 210 is medium effort valve, and I think pretty close to stock effort.
A 230 is very stiff valving and you pretty much have to have strong arms to move the steering wheel a lot. The NASCAR guys use these on super speedways where there is very little steering movement going on, so the car doesn't wander.
It sounds like, if your box is not a stock factory Saginaw box or has been changed out before, it might have a 185 valve in it, but I'm betting your stock box is about to take a dump on you...? Be carefull!
I hope this helps some, but take it for face value as this is just my opinion from past experiences... Jim