I set the desired temp on the termostat ( 70 F ) and the furnace keeps going until we get to 90F. It blows hot no matter what... It eventually kicks off but sometimes it takes hours...

Brand and model of furnace please. Thank you.

Brand : Goodman Model number CAPF4860C6BA

Your model number is inside the furnace... that happens to be the model number to the indoor coil that sits on top of the furnace for your A/C. Try this in the mean time... remove the stat --> jumper the red and white wires to call for heat --> let furnace cycle a bit and then un jumper the wires. Let me know if the furnace cuts off. I think your stat is bad.

My York furnace just keeps on running after the set temp

BigStew, try the same as I suggested to Charlie.