Refrigerators should be on their own dedicated circuit.

My guess is that, as your fridge has gotten older, it is drawing more power when the compressor kicks on that it used to.

It is also entirely possible that the fridge compressor has developed a legit ground fault that needs to be addressed.

The first step is unplugging or shutting off everything else in that circuit and leave it off to see if the fridge alone will trip the GFI.

One other question: are you plugging anything at all into the GFI outlet? How about the switched outlet?