I'm jealous, Kendall. That's quite a car.
Portland is a big enough town that there should be someplace that still sells radio tubes and may have a tube tester. Pull out all the tubes - remembering which one went there - and clean all traces of dust off the radio. Replace any tubes that test bad or weak - most are probably OK. If you can't find a tester, look to see if any tubes don't glow red, and replace those (sometimes, tube can glow and still be bad, however).
I think that tube-style car radios also had a plug-in vibrator that tended to burn out. See if you can test that.
This Google Search should lead you to some good information. This page tells how to repair radios from Chrysler Imperials, which shouldn't be much different from your Plymouth.

When I was about eleven, my hobby was collecting and fixing radios. I was very proud of the huge old Zenith shortwave I had, and I was thrilled to listen to Radio Moscow and Radio Havana Cuba. I was a "regular" at the drugstore tube tester.

-Joe-