Like brainanator suggested, measure between one of the outer legs to the middle leg and see if the resistance changes smoothly as you rotate the pot. Might want to do the same except with the other outer leg and the middle leg (resistance will change opposite). Go slow when adjusting/measuring though because it may be difficult to see irregular changes on a meter. The pot could simply be "noisy", which may not be visible on a meter; this could cause havoc with the radio electronics too.
That pot looks like a fairly standard part. Even mounts using the shaft collar. Depending on the value, a replacement
may be available at RadioShack. To get the value, simply measure the resistance across the two outer legs (the middle is the "wiper"). If RS does have the matching value, make sure it is "linear taper",
not "audio taper" (the audio taper resistance changes exponentially to compensate how humans perceive sound). The power rating is irrelevant for what it is being used for as long as the replacement is close to the same size (no PCB mount pots
).
You might also want to see how much extra room (if any) is available around the pot once installed in the radio in case the replacement is physically larger.
Then, it's just a matter of swapping the pots. You could eliminate extra points of potential failure by cutting the connector off that pigtail in your pic and soldering the wires directly to the pot - just make sure you know which wire goes to which pot lead.
Honestly, how hard would it be for manufacturers to use a rotary encoder device instead of a pot? This unit is inexpensive, so I can understand using standard parts, but some of the radios we use are rather expensive and you'd think they would use something that would last a bit longer.