I guess at this point, working without a schematic diagram, I would try substituting the good one for the bad or vice-versa, make sure you're not
on a wild goose chase and that the "bad" contactor is truly bad. Then if you are 100% sure the contactor needs replacing, start looking for a close match online, possibly from China via Aliexpress or Ebay.
It's normal for 120 volt ac contactors to buzz somewhat, they are not silent like dc types, but the buzzing doesn't necessarily mean a bad unit
If the other one doesn't buzz at all there's still something odd going on, and I would want to find out what it is before ordering any parts