Wouldn't a high load cause a large decrease in the core flux, since the secondary's current creates an opposing magnetic field that cancels out the primary's?A resistive load on the secondary, up to and including a short, will not increase the flux in the core. In fact, drawing current from the secondary will slightly decrease the core flux due to the IR drop in the primary.
Wikipedia: "On transformers connected to long, overhead power transmission lines, induced currents due to geomagnetic disturbances during solar storms can cause saturation of the core and operation of transformer protection devices."Presumably power supply authorities do their best to remove any DC offset from the mains supply. Considering that most consumers are connected to sub-stations with transformers it's unlikely under normal circumstances that a DC offset in the mains supply would be observed or if it did exist, it could not propagate very far down the distribution system. The next transformer lower down in the system network would not pass the DC onwards.