Pull the back panel out and clean the jellyroll condenser if dirty. If still not right, pull the back freezer panel and check the frost pattern, should be uniformly frosted, not partially frosted.
If ok, you probably have a refrigerant leak for a pro to look at.

All the symptoms point to a refrigeration system maxed out, that's why it is running all the time, new motherboard would not fix it.
For a fridg running all the time, the freezer should be colder. Also, depending on electric rates, you are probably wasting $100 per month.
I would try to replace the two fresh food thermistors, one located behind the damper cover, the other, ( if you have a second one) at the bottom of the fridg section, just might fix your problem. Stuck in ice water, they should each read 16.6 kohm.

Thank you so much for this post. I was able to troubleshoot and order part with your advice. The inverter was the problem. Only thing my husband and I did wrong was we turned the refrigerator off with the circuit breaker . After replacing part and turning refrigerator back on nothing seemed to change. We pulled refrigerator out from the cabinet, unplugged from the wall, and twenty minutes later plugged it back in. It worked!!!!!!!!! Thanks again your post helped us to troubleshoot.

Thanks for the heads up on freezer. Great idea for power surge cord. We moved our built in by slowly wiggling it back and forth. So heavy. It's been almost 24hours and frig seems to be working great. I was worried about compressor heat but read that was normal. Warm to hot touch ok. Super hot to the point where your hand can't take heat is different. Love these blogs-have saved a great deal of money troubleshooting so many different things. Thank you again and keep me posted.
K

Adding an update - this Monogram 48" side by side has problems that mainly come down to poor ventilation in the compressor compartment. The top door/flap that hides the mechanicals doesn't allow enough air flow. The compressor is constantly hot. The other week we were out of town so I turned our a/c up to 78. When we came home I noticed the compressor was crazy hot and the freezer was at 3 degrees - instead of its normal 0. It was running nonstop and struggled to get back down to 0.

I did some experimenting by testing the compressor compartment side temp - it was about 120 degrees. I left the mechanical door opened and placed a fan shooting air in the left side of the machanical area so it would blow through the coils and be pulled through by the condenser fan. The next morning when I woke up I checked the compressor and it was cold/room temp - I was worried that the inverter or something broke. The freezer was at -1 and the fridge was at 37 - so it was working right. I soon realized that with better ventilation the compressor will shut down and it appears that the efficiency is tremendously improved. I also noticed if you just leave the mechanical door opened it also helps the compressor stays cooler - but for obvious reasons you can't leave it opened. I bought a fan controller and two 4" fans. I use the controller to turn the fans on at a certain temp. I put one fan on the left side to bring in cool air and the other fan will be on the condenser side to exhaust hot air. I wish I would have thought of this years ago. You could also just have a small good fan set on the left side permanently blowing air. It's been a few days now and the compressor iis often off or just warm to the touch. Without these fans it was almost always running and usually really hot.

Let me know if there is any downside to what I've done. It seems like GE could have cheaply improved the ventilation on their own and saved owners a lot of problems.