I have a BT module (HC-05) connected to a UNO R3 powered by ~8v.
I needed to add more voltage to the Arduino, so I powered it with a little less than 12v and for some reason, it started to act by itself without any sense (it's a little car controlled by BT) and it didn't let me connect from the mobile application and later I think it burned out and so I had to replace it.
The Rx and Tx pins are connected to pins 10 and 11 (with the SerialSoftware).
When running at 8v it works perfectly.
What all besides the HC05 is connected to the 5V rail?
The on board 5V regulator is not heat sinked so will supply limited current before it overheats. If you are lucky the regulator will shut itself down, if not it will die.
Most likely what you really need is more current capability. Raising the input voltage to the on board regulator is actually counter-productive as the more voltage to the regulator the less current that it will supply before shutting down.
If you want more current on the 5V rail, connect the 12V to a 12V to 5V DC-DC converter (buck converter) and connect the 5V output to the Arduino 5V pin.
Power the motors separately. They generate electrical noise that plays havoc with the Arduino.
If you must use the same power supply for motors and the Arduino, don't use the Vin or "raw" power input. Add a separate 5V switching regulator, like this one, and apply power to the Arduino via the 5V pin.
I have no documentation on that. It is the ELEGOO SmartCar-Shield-V1.1.
And the motor driver it carries is TB6612FNG. Which supports even higher voltage