I have the same exact issue, it never made this noise until I received FSD beta. I noticed the buzz sound is gone when I first get into the car and when I need to use the turn signals the sound appears and does not go away until the car goes back to sleep. I also noticed the two stage turn signal option has gone away with this latest update. I can no longer lightly press the turn signal to activate the them, just hard presses work.
I also have another issue with FSD beta regarding turn signals. Whenever FSD beta engages a turn signal and I manual disengage autopilot, the turn signal stays on regardless if I make a turn or press the turn signal to turn it off. I actually I have to press the left signal and then the right signal to turn them off, which makes me feel like an idiot when people are behind me.