Blower motor turns on by itself
Older thermostats have a &quot;Fan&quot; switch that works independently of the system's On/Off switch.  That switch should be on Auto, not On or Circ or any other setting.
Newer thermostats have fan control too but some don't have a separate switch, just a setting somewhere.  Find it.  Make sure it's on Auto. Some newer thermostats have additional hidden &quot;installer&quot; settings that control the fan and limit what the user can do.  You can access those settings if you know how.
A thermostat can be wired incorrectly so the fan is always on.
The air handler can be wired incorrectly so the fan is always on.
Is it the thermostat?
If there is a fan switch, turn it to Auto (not On).  Did that fix it?  Good.
If it's on auto:  Turn off the breaker to the system.  Remove the thermostat from the wall.  Make note of how all the wires are connected to it.   Are any two wires connected to the same terminal?  That might be the problem.  Post a picture of that, and of the other end of this cable at the air handler.  If no two wires are on the same terminal, disconnect all the wires.  No wire should be touching the thermostat or any other wire or anything at all.  Turn the power back on.  Is the fan running?  If not, the thermostat is the problem.  If the fan is still running the problem is at the air handler.  Turn the breaker back off and put everything back. (You made note of where the wires were, right?)
Some thermostats are held to a base by plastic clips and can be yanked off the wall.  If you know you have one of these you can run the above test without turning off the power or disconnecting wires just by pulling the thermostat off the wall.  As long as you see that, on the base, no two wires are attached to the same terminal.