Yes, and the transformer temperature should rise if it has shorted turns. Also check the continuity of the HV winding.
What is the current draw?
A single shorted turn in a transformer always leads to a catastrophic failure. All other windings will be seen by their respective circuits as short circuits.
If the power supply is capable of producing the nominal voltages, there are no shorted turns.
The overheating of the transformer may indicate some overload.
Since the OP didn't deign to declare the model of the scope, we cannot determine its age. Nor have a more comprehensive view of the schematic. But since he said its old and analog, I'll suppose its more than 30 years old. In that case, the usual suspects are capacitors. All of them. Especially the high voltage ones. Not only the electrolytic. This followed by transistors and diodes. Resistors should be checked for drifted values.