New two bladed aluminum fan and water pump rebuild kit.

[Marshall V. Daut]

One way or another to fix this problem, you are going to need to remove the water pump and perform a minor pump disassembly to remove the shaft, even if you decide to install a collar to control shaft movement. The reason the shaft is moving back and forth is because the original design called for the inside end of the shaft to ride against a dedicated recessed flat spot inside the cylinder head. This kept the shaft from moving in and out - for a while. Because the shaft is made of harder material than the cast iron cylinder head, that flat spot will wear down, eventually allowing the shaft to start fore and aft movement.
Short of installing a longer shaft to compensate for this wear or installing a new head, a simple trick is to move the impeller farther up the shaft, meaning exposing more of the shaft in the back. This extended shaft portion in the rear will then ride again against that flat spot inside the head and keep the shaft from moving back and forth. To do this, turn the fan's castle nut backwards on the shaft after disassembly and run it up the threads until it is flush with the end of the shaft. This will keep the end of the shaft from mushrooming when you drive the impeller with a hammer. Choose a socket that is large enough to comfortably fit over the nub of the shaft in back, yet whose rim will conform to the raised portion of the impeller. Holding the shaft vertically with the front end of the shaft on a vise or solid metal surface, tap the socket so that the impeller is driven downwards (i.e., towards the front of the shaft). You will be able to see how far it moves when you lift the socket because of the different color of the previously unexposed shaft portion. Put the shaft back into the water pump body and test fit the pump without a gasket between it and the cylinder head. Tighten the four nuts. If you can easily turn the shaft by hand, the impeller needs to be driven down farther. If the shaft cannot be turned by hand or is very stiff, the impeller is down too far and the end of the shaft is sticking back too far. Move it towards the back of the shaft by placing the shaft upside down in open vise jaws and tapping the rear end of the shaft downwards. The vise jaws will spread the force over the face of the impeller so that it won't shatter or break. Just use the hammer only enough to move the impeller and maybe wear safety glasses, just in case the impeller is rusty or a crack is hidden. It won't take much to move the impeller up and down the shaft with a heavy hammer. Keep tapping and test fitting the pump until you feel a drag between the back of the shaft and the cylinder head's flat spot. The gasket will then provide the necessary clearance so that the shaft will not bind. Once you achieve the correct impeller placement, you could "stake" it in place by drilling a hole in the impeller and into the shaft, threading both for a small Allen head plug. Of course, if you worry about having to make this adjustment again in a few thousand miles, don't drill and tap anything. The impeller MAY stay in place. (I always stake the impeller, though) Once the pump has been reassembled, be sure to adjust the packing nut! The little distance you move the impeller will not substantially affect putting the castle nut back in place on the front of the shaft and installing the cotter pin. If it does, however, just dress a little off the backside of the nut until a cotter pin can be inserted through the nut and hole in the shaft.
This procedure may cause some readers to howl in protest, but I have found in 55+ years of messing with these cars that this trick works. It's FAR better than allowing the shaft to work back and forth while the engine is running, with the possibility of the impeller working itself off the rear of the shaft, allowing the fan blade to fly into the radiator core. Or you could just buy the longer shaft offered by vendors and dress the end of the shaft to achieve a slight drag. Lots of work and trial fitting required going this route, though.
Marshall