I would place the thermometer closer to the front of the fridge and on a middle shelf. The temperature should be much lower than 37 that close to the evaporator.
Generally speaking, this type of condensation is an indication that it's not cold enough. Presuming you have a single cold control, if the fridge section is too warm, the freezer probably is too. In situations like this, adjust the cold control for an ideal 37 in the center of the fridge. The best way to measure this is with a thermometer stuck in a cup of water. The freezer will be what it will be, but should maintain around 0 if the fridge is at 37.
Is this the same fridge that has the sagging freezer door? If so, frost accumulation as a result of the imperfect fit could be melting and finding its way to the ceiling of the fresh food section during periods when the compressor isn't running. Just a thought, although I think it would be obvious if the ill-fitting door was the source of the condensate.