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Hello, the Rc and Rh have a jumper between them in most cases and the Red wire from the furnace connects to only one of them, so what you describe so far is normal.
Tell me if the heating failed so you decided to replace the thermostat? if that is the case the problem would most likely be inside the furnace itself.
Tell me what color of wire connected to each terminal on the old thermostat,
Red to R
Yellow to Y
White to W
Green to G
blue to C
we can go from there