The thermometer being off a degree or two is no big surprise.
If the battery that you changed was a Coin type battery, then it is used to hold programming when the power is Off.
You may want to check and make sure that the thermostat is programmed for the type of system that you have.
If the battery went dead then most likely the Type went back to 0, the default.
0 is fine, if that is your System Type, If not then you need to reset it.