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You can tell if this is the case by AC on Blower running and you remove the cool or high speed wire on the board and the blower still runs then it is getting power from the welded bad contact.

Also be sure to fix before going to AC because if the heat contact is stuck and the cool contact is closed you are trying to run furnace on high blower speed and med or low speed and will cause...

The problem sounds like welded contacts on the control board on the heat relay. After so many contacts they arc and finally weld together or at least stick. Thump on the heat relay will sometimes...