Now: Is this just because I (probably) soldered them a bit weirdly, or is it because potentiometers in general are less robust than encoders? Should I just go for broke and get the encoders anyway?
While I cannot comment on your soldering (since I haven't seen it), I'd think that your problems stems from using common carbon track
potentiometers, which can have different issues in your application (besides a short life and a general low reliability).
If you switch to conductive plastic
potentiometers, you shouldn't have any issues, but they're quite expensive compared to carbon track types.
An intermediate quality is ceramic
potentiometers (can even be found with PTFE coating) and with potentiometers, both price and longevity usually follows the price.
One HUGE advantage to pots is the ability to have absolute positioning. The encoders are all relative.
... Except for the absolute position encoders
i wonder: can the Axon be programmed to store a small file in its flash memory that stores the encoder readings once the axon is shut off?
It has got 4kB of EEPROM and that's where you would usually store such info.
However... A better solution is to start by fully contracting (or extending) all controllable joints to whatever limit switches you may use for a known start up, as storing a position that might be moved by external force when shut down will lead to errors (or worse).
Only ever store and rely on info that you are sure cannot be altered by other means.
Whether you should be using potentiometers or encoders will depend on the size restraints and your budget - If the budget's large enough, you can have whatever size and form you like.Here's another possible solution
from France, that might be worth taking a look at.