This is just a guess, but it's possible those lines could feel like they're at the right temperature but there's not enough freon in the system to bring the fridge down to the proper temp. The only way to tell is to vacuum out the freon and weigh it. If it doesn't have the proper charge in weight there's a leak some where and that can be a real problem. The factory label often lists the proper freon charge in weight, but it takes special equipment to vacuum the system, weigh it and refill it with the proper charge. And even if you do this there's no guarantee that it won't leak out again. They have freon detectors that sence leaks, but you have to have access to all the coils to use the detector. They can also put dye in the system to look for leaks visually. At least they do this in cars. Not sure about fridges.
The use of R12 is banned in some instances and is expensive. Explore all other remedies before going this route. Also consider that this is a closed system. Once it's opened it's often hard to keep it from leaking again. I share your want for not wanting to give up on your fridge, but some times, especially if it has this problem, there's not much of a way to save it.