The homeowner should be shopping for a new service panel, even before a new dryer. Around here (western PA), banks won't finance, and I don't think insurers will cover a structure with a fused service panel.
Motors can develop insulation current leakage which a DMM resistance test will not indicate. A motor with suspected leakage needs to be checked at operating voltage (or better yet, Hi-pot tested with a Megger-type tester). Contamination inside the motor can cause leakage paths.
It's likely that something else may be involved in the overall scenario. As you suggested, a 120V motor probably wouldn't take out both fuses.
Fuse box hardware generally always loosens with age. Vibration (even though it's not felt continuously) and thermal cycling will lead to loose fasteners.. especially when many fused panels are over 40 years old. Loose fasteners anywhere in a circuit will increase the resistance in circuits, and usually break down from excess heat.
A qualified, experienced and knowledgeable electrician should replace the service panel.. if a homeowner fits that description, that person could probably perform the installation in a couple of hours.