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A transformer with a shorted secondary CANNOT be used in the circuit because the short is INSIDE the transformer and cannot be cleared by merely disconnecting the loads from the affected winding. If the short is in one of the filament windings, then the winding may be removed and re-wound since the filament windings are the outermost windings. Power transformer anatomy follows some simple conventions. The 120V primary winding is almost always the innermost winding, followed by the HV secondary winding, then the lower-voltage secondaries (bias windings - common in guitar and hi-fi amp transformers), then the filament windings. Rewinding an entire transformer is feasible, but it is quite time-consuming and very labor intensive. It involves taking the core apart and building a jig (rotating block with a crank - like a simplified fishing reel) to hold the windings. Slit the layers of paper, unwind and count the turns of each winding, take notes and make sketches. Obtain magnet wire of the proper sizes to match the original windings and use paper like the original (heavy Kraft paper) to separate the layers of windings. Neatness counts and it takes practice. Refer to "Construction Practices and Data Tables" section of an old copy of The Radio Amateur's Handbook. I have the 1975 edition which still featured a fair amount of tube data. I hope this helps.
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