[morzh]

The simplest check is, put the ohmmeter across the AC plug, while Off. Turn it on. The resistance should be on the order of a few hundred ohms or less . If very high (Kohms, Mohms), your filaments are open or the tube sockets do not make good contact.

Simplest test first...

I have an "auto range" multimeter that automatically sets the range ( I can also set it to manual range if I wish).

If you mean from spade to spade...when I try this it starts out with a 0 reading and when I turn on the power switch, the display cycle up through a series of Mohm readings 18...19...then returns to 0. 

Thing is it doesn't to it every time I turn the switch off then back on again and switching to "manual" and testing various ranges give me 0.

If I test from the + spade to chassis I get 0 when "off" and .474 mohm when "on".
 

This does not make sense. What you are telling me is you have a short circuit between the plug terminals whether Off or On.