I have just replaced my thermostat and now furnace blows cool air (fan on constantly), even with thermostat control in off setting. Followed all install directions w/ new thermostat
To the letter. Is this a limit switch malfunction or what?

Disconnect all of the stat wires and start from scratch. I know you said you followed the directions to the letter but I've made a few silly errors doing wiring. Is this a four wire stat controlling a furnace with A/C?

If it was working right before changing the thermostat, I would go to the sticky at the top of the forum and find the section on thermostat wiring. Also check if the wires follow the regular convention, i.e. white wire on W, at the furnace. You want the same wire on W at both the furnace and thermostat, preferably the white wire.