A number of things would cause a transformer to run hot; if you load it continuous 95-100% which is not good practice, solution-install bigger transformer if cannot permanently reduce load. Transformer winding temperature is measured indirectly unless your transformer has fiber optic sensors embedded in the winding.
Never re-energize a transformer that has tripped on Over Pressure (Dev 63) without first performing IEEE routine test on the transformer. Most oil filled transformers over 1,000 kVA (mineral oil filled) are dual rated for temperature rise, 55C/65C. If the oil temperature is approaching 55C you still have room. If you are seeing quick spikes in temperature most likely this is a bad sensor/transducer/wiring but it is possible you have an intermittent short circuit. If this is the case you should see large current spikes say over 1000 amps. You may want to take a gas sample and perform a DGA (IEEE Std C57-104). This will tell you if you have arcing in oil. For an internal fault a rapid rise relay or sudden gas pressure device (IEEE Dev 63) that trips the primary breaker is common.
Indirect measurement has two components, one is direct measurement of top oil by a sensor or a probe and other is measurement of winding current. A current transformer is placed on one of the three phases of untapped winding. This current transformer feeds current proportional to the load current in to a heater coil in the winding temperature gauge to provide additional deflection. This additional deflection corresponds to estimated or tested winding gradient. Additions of these two deflections, one of top oil sensor and other based on current transformer secondary current, provide the estimated winding temperature. So the accuracy of indicated winding temperature depends on the accuracy of current that is fed through the heater coil of temperature gauge.
So please check the load on transformer if it is close to its top rating, if so, try to reduce the load.
Also check what the reading of oil temperature gauge is. if it substantially lower than winding temperature, around more than 20 deg lower, than it is possible that current that is fed to the heater coil is not accurate.
So I would suggest to check if load current on all 3 phases on both primary and secondary side is almost equal, compare them with the name plate current to see if one of the phase is drawing much higher current. As for tripping, trip on load side first and then if primary current is still high, trip on primary side also.