Even if it's plugged in and cold, it could still have a failed auto-defrost system. Based on my experience, this is a common failure mode on fridges, isn't too hard to diagnose and repair, but often results in people "Eh, it's 7+ years old, they just don't make things like they used to, better buy a new one" and doing just that.
You could specifically test that on a fridge that's running by figuring out how to trip the timer that cuts off the compressor and engages the defrost heater. Then, listen for melting in the freezer area and ensure that the timer self switches back to cooling mode. Easy to do, but might take you 30 minutes, so hard to pull off in a typical Craigslist hit-n-run type of purchase. In most cases, a repair to the defrost is going to take 45-75 minutes two times (once to take apart and find the part that's bad, then put it back together and use it while it slowly ices up inside and the parts come in, then again to fix it). The parts are cheap ($20-60) and the system is simple, typically having only 3 parts.
The other thing to think about is that fridges over 10 years old aren't going to be as efficient as modern fridges, so at some point, the purchase price plus energy cost will make a modern fridge a better buy if you're paying both. If your buying a fridge for a rental where the tenant pays the electricity, that math doesn't apply (though the principle applies).
When you transport the fridge, if you lay it down to transport, stand it back upright for at least an hour (I'd do overnight) before plugging in. This might be an old wives' tale/superstition, but I've heard it fairly often.
