Filter 'lytics going dry is the most common cause of hum, but there is a wide variety of other causes, some of which can be bizarre indeed.
I had a Zenith AM/FM table radio in the barn here which developed an elevated hum level. Also the dial drive had suddenly failed. Turned out a mouse had gotten into the set, bit the dial cord in two, and then sauntered over to the ouchput tube (35C5) and peed there. Such act turned out to be his last, because there he lay--dead. I had to restring the dial drive and replace the contaminated bakelite wafer tube socket. I treated the radio to a nice porcelain one instead, as that substance is designed to handle pee.
If you do decide to recap you should indeed do all of them, as suggested. Also most of the Zenith AM/FM sets of the time used a selenium rectifier. You can get a little more spirit out of your radio if you replace that as well, as those do fail, and when they do they emit fumes that smell as if someone broke wind. And those fumes are hazardous to inhale. I usually replace them, using a 2.5A 1000PIV diode in series with a 22 ohm resistor. And obviously one needs to observe polarity.
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