[Arran]

The dim bulb test will not indicate a shorted transformer per se, but will only tell you if there is a short in the radio. I should point out that even if everything is fine the bulb will still glow, it's more a question of how much and for how long. The fact that the dial lamps glow is an indication that the filament winding should be fine, the bulb acts as a ballast to drop the input voltage, as a result the tube filaments may be too dim to see at first, it will take longer for them to warm up verses the pilot lights. If you have a light bulb in series with the line, you turn the set on, and it glows brightly and stays that way, then there is a short in the power supply section. At this point I would take an ohm meter, with the set unplugged of course, and check each winding, I would also check between the rectifier tube filament and the B-, you could simply check between the positive and negative sides on the filter condensers as well, low or no resistance indicates a problem.
Best Regards
Arran