I would seriously question the integrity of the amplifier if the case temperature was so hot you can't hold your hand on it. If the temp is that hot, the varnish type insulation in the transformer core will degrade.
If your bias current is off by a significant amount, then it would increase the heat loss in the transformer since it is driving more power. However, it isn't likely it can be off so much the transformer will become so hot you can't touch it.
New filter capacitors will not increase heat dissipation in a transformer but old ones that leak will. If all this started when you replaced your caps, you should verify they are not connected backwards, as polarized caps installed backwards will indeed leak significantly. If this is the case, then you should replace the caps - don't just flip them around. You are asking for trouble if you do this.
It is possible if the transformers ran at a high temperature long enough, they could be damaged to the point of requiring a rebuild. High heat is not a good thing, ever, even in vacuum tube equipment. Granted vacuum tubes have cathode heaters but even they have an optimum temperature and excessive heat will destroy them.
Generally speaking, you can't hold your hand on a properly operating vacuum tube, but you can on the transformers. If you can't, something is seriously wrong.