Now for the cure - I put an R substitution box in series and increased the resistance until the oscillation stopped. Then I read the value: 15KΩ. So that was the minimum series value to increase R by. I then got a 100KΩ pot and jumpered it up in an L-pad configuration, and wound the pot until the volume was similar from the 78 and radio inputs. Reading the pot values on the multimeter gave 66K in series and 33K for the shunt path. I selected the closest values (68K and 33K) from my 1% metal oxide resistor draw and soldered them in on the RCA plugs of the 78 record deck.
Job done - now back to listening to Bing, Elvis, and Doris Day.