[Arran]

 Just to clarify things, the power transformer is DOA in your 38-12, not the output transformer? Is the field coil on the speaker fine as well? Yes, I think that hitting the upcoming swap meet might be the best solution for a replacement, given that a 38-12 was a popular set, and that there were later versions of it and the 38-14 that had Bakelite cabinets, you may be able to find a parts chassis with an original power transformer. If not there are a whole host of different makes and models that used power transformers of a similar size and style that used either a #41 and a #84 or a 6K6GT and a 6X5GT or a 6X4 and a 6AQ5 (no not every 6X5 equipped set was badly designed like the brand Z sets) , but a some even had a proper 5 volt rectifier winding.
Regards
Arran