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No expert here but Here is a way to test your transformer. Remove the 5u4. Mark all the wires going into the transformer and disconnect them. Run the ac line from your voltage splitter and light bulb to the primary winding wires and turn on. See if you get a short still. If no short, Hook up the secondary winding wires one at a time. { I do the heaters first} That may lead you to the problem. Also, Look for wax caps on b+ lines.
It does sound like the transformer is bad from what you have been writing. That weird smell and the hot cord. B Careful, A bad transformer can make the chassis go to the hot side of the ac line. The other guys are right, Get rid of all the old electrolytic caps.
ED