Quick and simple test:
Measure the primary resistance when the transformer is at room temperature. Then apply a load to it and allow the transformer temperature to stablize (about 45 minutes). Now disconnect power and measure the primary resistance again. As long as the hot resistance is less than 1.27 times the cold resistance, you should be within its design limits. Obviously you should start with a modest load on the secondary, and then you can slowly increase it until you reach the 1.27x hot temperature resistance limit.
The theory behind this is that the resistivity of the copper winding changes according to a well defined coefficient, and assuming the room temperature resistance reading is done at 20°C, then 1.27 times that resistance reading corresponds to 90°C which should be a safe operating temperature for virtually any modern transformer. This method also has the advantage that you're measuring the average internal temperature which will be higher than the surface temperature.