Blower motor continues to run after turning off the vehicle?

How long does it stay on? Did this just start when cold weather started? Are you using defrost? Asking because GM has an after blow on many vehicles. It runs the blower motor to dry the HVAC case and cores when it is humid or when the AC has done. Your AC runs when defrost is engaged so the moisture in the air doesn't turn to ice on the inside of the glass. This is enabled or disabled along with the run time using a scan tool. Normally it runs at whatever speed the fan was on when the key is turned off. If it turns off after 5-10 minutes, it is likely the after blow. If it stays on longer than that then it is a concern. If you remove the fuse overnight or however long, does the blower start up again as soon as the fuse is reinstalled, even if the key is off? Do you have the full automatic HVAC that you set to a temperature, or does it have the dial type controls?

They are designed to run even after the key is off and removed. That is what the after blow system does. It keeps the blower motor running for a while after you shut off the vehicle so it dries out the HVAC system to prevent mold growth in the system from the moisture. Most are set to run between 30 seconds and one minute in the newer versions depending on the temp and humidity. That is why I asked how long it runs after you turn the key off. If it keeps running for longer than 5 minutes then there is a problem. The only way to shut that feature off is to use a scan tool to access the HVAC control module and see if it is active and to shut it off if you wish.
It also can't be the relay if the system works normally once you put the fuse back in. If it was a stuck relay the blower would turn back on as soon as the fuse was put back in. Try this, run the vehicle and when you shut it off time how long the blower stays on. The normal software system runs it 30 seconds, then off for a minute, then on again for 30 seconds and it repeats that until the system determines the cores to be dry. There was also an add on module that had it's own settings for the after blow but those work differently and keep the blower on longer.
What I would do is grab a scan tool and go into the HVAC and look at the after blow settings and see if they are active, then if the blower is staying on longer than it shows in the settings, I would disable it in the system. Does the blower still stay on after it is disabled? If yes then we need to look at the blower motor control module located in the HVAC case near the blower motor. If the blower is on then you would check the Purple wire with the white stripe for a pulse width control signal (you can use a test light with a needle pushed into the rear of the connector, if the blower is running on high it would be a bright light, it would get dimmer for lower speeds and off if there is no signal) If there is still a signal to the module when the scan tool shows it to be off the control module in the dash is faulty. If however there is no on signal you would change out the blower control module itself.