Frigidaire Ice Maker Leaks Water?

My side by side Frigidaire refrigerator is 12 years old and has recently started to leak water. When the water dispenser is used for a couple of minutes, water starts to leak and drip from the ice maker into the ice tray/freezer. The ice maker makes ice as normal and only leaks when the water dispenser is used.

I tried replacing the ice maker, but that didn't fix the issue. Any other suggestions? Does this sound like a water inlet valve issue or not really since the ice cubes are normal size, are made at a normal rate, etc., and the issue only occurs in relation to extended use of the water dispenser?

I would like to understand all of the things it could be in order to determine if it's worth exploring/troubleshooting further or better to just replace the entire fridge at this time, given it's getting up there in age.

I'm hesitant to replace the water valve unless the consensus is that it will resolve the issue. Reason being is the tube connections on the latest valve model are different than those of the original, and so tube modifications would be required.

Disclaimers: I plan on moving within the next 3-4 years. I'm not super handy, but can follow directions well. (I had no problem removing, sourcing the part, replacing the ice maker.) I have no experience working with water lines or the innards of appliances. I do have experience with computer/technology repairs.

Thanks in advance for the assistance!