Freezer and Fridge stop working

The freezer in our GE Profile approx. 10 year old fridge began to go into long defrost periods and the ice in the receptacle would meld into one large lump. The first 3 repairmen replaced the icemaker. The problem kept happening and actually worsening as the meltdown periods became longer and longer and affected the coldness in the fridge side as well. After some research on the internet, we read that the thermistor should probably be replaced. This was done and it seemed to fix the problem for a couple of weeks. Then it started to happen again, the repairman came and checked everything he could, plugged it back in and it reset itself and came back on. We ordered another thermistor which was brand new as we thought perhaps the previous one might've been used or old as the mechanic had brought it, not in its original package. After it was changed, freezer/fridge worked again briefly, 4 or 5 days. After that, when it went off, we resorted to reset it by unplugging and replugging, this sometimes worked. The last time as we pulled it out, we heard a brief sizzle sound and saw a light behind the panel; we realized something had burned out. The repairman came and saw that the starter had burned itself out. He replaced it and after 40 minutes the freezer and fridge were back on. We hoped that perhaps the starter had been the problem the entire time and maybe as it was dying and before burning out completely it was the cause of our freezer and fridge going off. No such luck, everything worked for 12 or so hours then off again. Would appreciate any insight/help before having to buy a new $1,500 fridge. It seems to me that it's something fixable as it works some times. if it was done wouldn't it just not work at all? Thanks for any help/tips you can give.