[RichG]

Hello,
Thanks for the replies, everyone! I feel a bit foolish for not thinking about how transformers are actually constructed (probably comes from never taking one apart).  In my defense, I did say rewinding transformers was not in my wheelhouse.  The explanations made perfect sense.  I feel better about looking for a replacement transformer, knowing I'm not limited to radios but can include TVs.  I did try to bring up the original transformer on a variac. When I got to around 20 V or so, the transformer started to hum and get warm, so I gave up. It is definitely toast (no pun intended).
Brad, I totally get finishing one project before moving on to the next!!
Thank you everyone1
Rich

[Arran]

With regard to the 39-116, (at least the U.S versions) I believe that they used rubber/gutta percha wire for the leads, which of course can rot or break down causing shorts to ground, much like the old A.K transformers can, so if someone powers one up without inspecting or replacing these leads beforehand, transformer failure is a possibility.
As for viable replacements it may also be worth looking through the selection of power transformers intended for guitar amps, such as what AES has available. You will have to find out the specs of the transformer in the 39-116, add up all of the currents of the tube heaters, rectifier winding is separate, and the voltage and current of the H.V winding.
Regards
Arran