Yes, one side or the other of the transformer has an open (burned wire) in it. Probably the primary (high-voltage) side.
If you want to check, use your multimeter on its resistance (ohms) setting to check the primary side. With the transformer unplugged, place the meter probes across the plug prongs. You should get some non-zero reading. If it's zero, the transformer's fried.
Some transformers have built-in fuses, in which case the fuse could be blown. It might be hidden under the tape, though. They're not really meant to be replaced; if they go, you usually need to replace the transformer (though I have replaced them, being a cheapskate).
Fortunately, transformers are easy to come by in today's world. Let us know your progress.