No. Spindles can be switched side for side as long as you switch the calipers too (so the bleeders remain on the top).
So, let me get this straight- if you turn the car to the right, it will keep turning right unless you manually turn it left to recenter it? Does it do that in both directions?
It could be the adjustment on the steering box. It should want to re-center itself, if its "sticking" it could need to be re-centered. Or the steering box could just be worn out. I'd still check the pitman, idler, tie rods etc to make sure nothing was loose or broken. You should also check the steering box mount on the K frame, they have been known to crack at the welds, and then the mount flexes back and forth so the steering box moves when you turn the wheel.