Testing a transformer
Your understanding of transformers is a bit off, but the root cause may be as you suspected...

A transformer (usually) consists of two coils of wire, a primary and a secondary, wrapped around a ferromagnetic core.  An AC voltage applied to one coil creates a changing magnetic field in the core, which induces another AC voltage in the other coil.  The ratio of the primary voltage to the secondary voltage is determined by the ratio of the number of turns in each coil (the turns ratio).  When AC is applied, the impedance across the primary coil is that of the load circuit multiplied by the square of the turns ratio.

An ohmmeter applies a known DC voltage to the circuit being tested and measures the current flowing into the circuit.  Because it's DC, the magnetic field in the core doesn't change, so there's no voltage induced in the secondary coil.  What you are measuring is simply the resistance of the primary coil, which should be low.  

Barring some unusual circumstance like an electrical storm causing a surge or a teenage son turning everything up to 11, you probably have a problem somewhere in your system.  I think I would call in the service people at this point (I'm not familiar with your system, so don't know if it can be serviced by end-users), but if you're up to troubleshooting, and if you can disconnect or turn off everything on the secondary side of the transformer, you could try replacing the fuse and re-applying power.  If the fuse blows again, you know it's the transformer at fault.  If not, reconnect or turn on devices one-by-one until the problem manifests again.