Rheem gas furnace blower will not shut off
(Rheem Imperial 80 =)

After it is done heading, the blower continues.
I have set the thermostat to auto.
The batteries in the thermostat are fresh.
I actually bought a new fan limit switch and install it. Still continues to blow.

I can shut the furnace off with the switch on the furnace. But that is not
a good solution,

What part is broken on the furnace?

There could be a add on relay in your system for a A/C unit and if so the contacts could be welded together. The new fan and limit control if wired and set properly should have taken care of that problem with the exception if it has a add on relay.

Just thinking out loud. If you replaced the fan limit switch. Some fan limit controls have a on and a auto switch on them do you think you may have it in the on position, if so it will run for ever.

If you turn off the furnace by the switch, then turn it back on after it cools... does the fan come on, or only after a heating cycle?