How can I test the OT with minimal test equipment?
Use the amp's own PT as a VAC source to measure the Pr:Sec VAC ratio on the OT
With the PT switched off,
Disconnect the OT primary and secondary winding ends from the circuit. Make sure the floating winding ends don't contact anything else (or each other)
Disconnect all the PT secondaries from the circuit. (and do same trick with ensuring no shorts to ground/other circuits etc)
1) connect the OT secondary winding to the PT heater winding. Clip a VMeter cross this connection (to measure VAC - you'll be looking for ~6VAC when you switch everything on)
2) Hook another couple of VMeters to the OT primary - you want one meter measuring across each half of the OT primary - you'll be looking for 100 VAC or so across each primary winding half.
Switch the PT on (after making sure the mains fuse etc is properly connected and there is no unwanted shorting of leads etc).
Measure the exact VAC across the OT secondary (to confirm the exact voltage (e.g. 6.3VAC etc)
Measure the exact VAC across each half of the OT primary
For a healthy OT, you're looking for the voltages across the OT primary halves to be the same - AND you're looking for the desired Pr:Sec VAC ratio (which will be the square root of the OT's designed Pr:sec impedance ratio)*. Note - for a PP OT, the component you're looking for (to get the Pr:Sec VAC ratio) is the end-to-end VAC across the whole OT primary winding (Edit: also note, this won't be a 1kHz signal, it'll only be 50 or 60Hz depending on the mains in your country - but this is close enough to give you a fairly accurate picture of the OT VAC ratio)
Switch the PT off before touching any leads.
*say you measure 200VAC across the OT primary end-to-end with 6.3VAC measured across the secondary. That's 200:6.3, which is 32:1. 32x 32 =1,024 so impedance ratio would be 1000:1, which is an 8k plate to plate reflected load with a 8R speaker connected to the secondary