Sounds like the compressor is running. The problem could be as simple as a bad fan motor for the evaporator, condenser, or both. I'm betting it's normal for the evaporator fan to cut out when the freezer drawer is opened, but that would be easy to confirm one way or the other.
If this fridge were near me, I'd at least take a look at it. It's just sooooo clean, and those swing-out shelves were a major improvement over the revolving ones.