Sedimant in hot water pipes
While the dissolved minerals in water can precipitate out of solution anywhere in a plumbing system, heat and/or reduction in pressure is typically responsible for almost all visible hard water mineral build-up or accretion(bicarbonates of calcium and magnesium) on plumbing system components.
When water goes from a pressurized state in a pipe to a depressurized state where it exits the pipe at a showerhead or faucet aerator, the water becomes agitiated and gases in the water such as CO2 are released from the water increasing the pH of the water. All of this causes the minerals to clump, come out of solution, and accrete (accumulate) on the the surfaces of fixtures, enclosures, showerheads, and faucet aerators.
All this being said, it is more likely that the problem you're experiencing is a result of the deterioration of the water heater dip tube. Given the age of the system, the water heater was probably manufactured with a plastic dip tube made by Perfection Corporation of Madison, Ohio. These dip tubes were not impervious to the chemicals and temperature conditions in water heaters and they deteriorated inside of the tank. The result (besides tank inefficiency) is that small pieces of the deteriorated dip tube plastic are carried through the hot water piping system and lodge in shower heads and faucet aerators. You can determine if this is the case by removing a clogged aerator and looking for small bits of white to brownish white material in the screen. Remove this material,allow it to dry, and then apply a flame to the material. If it melts, smokes, or burns, it's material from the dip tube.
If the above is the case, the procedure to follow is first to remove the old dip tube (it should show evidence of deterioration) and replace it. Then, follow the proper procedures for draining, flushing, and refilling the tank. Next, remove all showerheads and faucet aerators and open the hot water valves
to flush out the hot water lines. Once the lines are flushed out, close the hot water valves, clean the showerheads and faucet aerators of all particulate matter and reinstall them, and relight or restore power to the water heater.
I hope this helps. Let me know if the dip tube was deteriorated. Thanks