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The blue connector is to supply power to the dash cam. In that connector, there is a ground wire, a power wire when the car is on, a power wire when the car is off. The colors listed in an above post. The power wires to your dash cam can be connected to these wires. Many dash cams require two power wires- one when the car is on for normal continuous operation and one when the car is off for parking mode. All this power comes from the blue connector. There is no need for any other power connection.

What you did: It seems like you connected the always on power to the cigarette lighter plug (which is not powered when the car is off). This backfed power to the fuse block and as a result powered things on in the car that normally should be off. There should be no connection between the blue plug and the cigarette lighter plug. In fact, there should be no need for the cigarette lighter power for the dash cam if you are using the blue plug.