I restored a 28T962 with the same chassis about 12 years ago, and had a similar problem. The Zenith TV sets of the late 40s have the same IF transformer problems as their radios of the same time period. These are the transformers which have the silver-plated mica disc capacitor assembly in the base of the transformer, and they are well known for intermittent connections and leakage caused by migration of the silver on the disc. I repaired the defective transformer on mine by measuring the capacitance of one good part of the capacitor with my capacitor checker, and then I removed the disc and wired in new disc capacitors of the correct value.

Check all of the voltages on the tubes in the sound IF, detedtor, and audio circuits.