Refrigerator is aprox 16 years old and is a side by side.

My refrigerator is a side by side and does not seem to stop running. It is approximately 16 years old. I took off the panel in the freezer part and it was frosted up. I took a blow dryer and defrosted it. It worked for a couple of days (it shut off at times like it should), however now the food in the refrigerator part is getting frozen and again it is not shutting off. I was wondering what the problem would be, and if it is worth having a repairman come in to fix it?

First vacuum the back where the motor is, the fan, the coils if exposed - whatever needs it.
Check the gasket all the way around the doors.
Meanwhile shop for a good deal on a bottom freezer energy star rated fridge, because when I bought one, my electric bill dropped $35/mo and stayed that way, so it will pay for itself in 4 years (cost 1700). It's a 33" double door/1 drawer LG stainless and I love everything about it. Never thought I'd get excited about a fridge. Oh, and forget about ice cube makers and all that. Waste of money, space, and repairs. Freeze ice the old way.

I forgot to emphasize freezer on the bottom - that's where all the efficiency is. (And wait at least an hour or 2 for the fridge part to get cold when you first plug it in. I was on the phone saying 'freezes but fridge not cold' and the reps didn't know enough to tell me why I had to wait - the air from the freezer is transported up a channel at the back to cool the fridge.)

Are the coils under the unit clean? I bet they are filthy which will cause your problem. DrTom4444

The other possibility included is that your defrost system has failed in addition to having dirty coils. Remove the cover to your freezer coils and ohm out the defrost heater and if it is bad replace it and the defrost terminator. I would also replace the timer because it is so old. The timer is either under the unit or next to the t-stat in the refrigerator section. It is very simple. DrTom4444

Your defrost system has failed which is simple to fix. First, clean the coils under the unit, then remove the cover to the coils in freezer and check the defrost heater with an ohm meter which should read about 10 ohms. If bad replace the heater and terminator t-stat which is attached. Also, replace the defrost timer since it is so old. If you clean it well and replace these parts you will have a new refrigerator which will use much less energy than a new one. DrTom4444

Let me amend my answer: You have a defrost system problem.If it is a normal type it has a defrost timer which turns off the fan in freezer and the compressor and energizes the defrost heater until the terminator t-stat turns it off or the timer turns it off, whichever one is first. It's a very straightforward system that is easy to work on. The electronic ones are totally different, but normally when they go bad only the heater and terminator needs to be replaced. Just remove the cover to coils in freezer to access heater and terminator. Don't forget to clean the coils under unit, too. Many times the timer is under the unit near the coils or near the t-stat in the refrigerator section. Let unit thoroughly defrost before working on it. After removing the cover to coils you will see wires leading to a heater attached to the coils. On a side-by-side the heater is a glass rod with an element inside that you can see if it is bad; on a freezer on top model it is aluminum and wrapped around the coil that will burn apart often. There is a defrost t-stat attached by wire to one side of the heater and clipped onto the coils. It is round and has two wires, one to the heater and one to the power from defrost timer. With refrigerator unplugged turn timer until you hear it click and stop. Have wires unplugged to defrost heater and defrost t-stat and get the plug and the wire supplying power to heater. Put meter on ohms and put one lead on wire supplying power to heater and one to plug prong and if timer is good you will read about zero on both prongs – one to each wire supplying heater power. Then put one lead on each side of heater and if you have no continuity you need a new heater. Replace heater and terminator together and be very careful when replacing it, especially on the freezer on top model because if you bend the wrong thing you will break one of the delicate capillary tubes and lose all of the Freon which will ruin the unit. If the timer is bad you will need to replace it with one that matches. You can test it before you take anything apart by advancing it until the refrigerator turns off. Then see if the heater defrost the unit and then brings the unit back on; if it does not then all you may need is a new timer. DrTom4444