Wait, you cut the core?
Well, congratulations, you have ruined/severely damaged it.
Transformers are made of lots of sheets of steel, with very thin insulating layers between them. This is to keep eddy-current losses from causing lots of heating, as you have discovered.
Ferromagnetic materials are also good conductors and a core made from such a material also constitutes a single short-circuited turn throughout its entire length. Eddy currents therefore circulate within the core in a plane normal to the flux, and are responsible for resistive heating of the core material. The eddy current loss is a complex function of the square of supply frequency and inverse square of the material thickness. Eddy current losses can be reduced by making the core of a stack of plates electrically insulated from each other, rather than a solid block; all transformers operating at low frequencies use laminated or similar cores.
Microwave transformers are normally somewhat lossy, as they are not operated for a significant period of time. A stock microwave transformer will get noticeably warm if sitting unloaded for a while. You have just increased the losses by many times, by shorting out the laminations.
There is nothing you can do with the transformer you have. You need to get another transformer, and not cut the core to remove the secondary. You have to remove the secondary without damaging or dinging the core significantly, and then wind your new secondary in place. by threading the wire through the core.
For what it's worth, microwave transformers run pretty warm without any load. Have you compared this transformer to another, without the core damage?
I'd be interested in some measurements of no-load power draw on the hacked-up transformer vs a stock one. That would let you measure the increase in losses due to eddy currents.