The only thing that makes any sense to me is that the primary is shorted to the HV secondary. As Thomas said, the way to check it is to place one lead of the ohm meter on one of the AC primary leads and the other on one of the secondary leads. My bet is for the HV secondary so one of the rectifier plates. It should read open. The primary is supposed to be completely isolated. It should be open to the chassis and all other points in the circuit.

Instead of replacing the transformer, you could add an isolation transformer between the line and the primary of your transformer. I find transformers with dual primaries useful for this. One primary to the line and the other primary becomes a secondary. Or, you can try to find a transformer designed specifically for isolation. Of course, you have to find room to hide it in the set somewhere.

The isolation transformer must have a high enough power rating for the job. If the isolation transformer core is about the same thickness (or thicker) as the transformer on your set, then you should be good to go.

Tony