There are no "little parts on the end" of a cartridge which can be replaced separately, so you are on the right track, but may not be necessary to replace the pipes in the wall. MIXET, while defunct, was a widely sold model for a number of years, so parts are still readily available. Any Depot store, ACE hardware, etc. should have them. On line there are any number of plumbing parts suppliers. The HD dot com website is pretty worthless, though.
One thing you have to look for in Mixet, there are a short stem and a long one for pre- or post- 1968. The parts websites usually explain this with pictures. Also, there is a version of the newer stem in an -EXT model, a long shank for remodel situations where wall thickness is added.
There is also a company who makes a special retrofit cartridge to cure the cross-over problem common with Mixet. They are pricey, and if you do not have a pumped recirculating system, you don't really need this. If you do have a pump, them $50 for a stem will save you that much and more on your hot water bill.