Dishonest technician? A classic case

We have a 15-year-old, side-by-side GE refrigerator that has recently not been cooling well nor freezing properly. We had a technician from the store where we purchased it come out and he pronounced the compressor shot – said he ohm'd the motor and his tester indicated an open circuit. He further said the freezer compartment fan is not running, but he thought that is because it gets its power from the compressor motor. He told us our options are to replace the compressor (~$500 - $600 installed), or purchase a new refrigerator/freezer. We did not want to make an instant decision without a second opinion, so we'll likely call someone else to come out and look at it. However, even after the technician left, the freezer continues to freeze, although not rock solid. We put water in a tray and placed it in the freezer and only an hour or so later, ice had partially formed. We checked later and the ice had thawed, then again later, and it was mostly frozen again. Does this behavior make sense if the compressor is truly shot, or does it sound like it might be something (hopefully)less major and expensive? I just don't understand how any refrigeration, and particularly freezing, can be happening with no compression, especially in Houston's hot, humid climate. I'll appreciate any guidance you might offer!