You can connect a multimeter and read if you get good signal.
Yo can swap the relay.
Of course,should get a good battery.
Just some info about relays may point out to the issue.
The relay have a rated voltage to close the contacts providing 12v at coils. It has tolerances, you could turn on the relay with 9v there are other voltage,"retaining voltage" the minimum voltage to hold the relay on, it could be -%30, well if you battery fluctuate with the loads when the relays turn off for lack of voltage the battery voltage rise again and the relay turns on and off because enters at resonance state.
Relay turn on voltage
Relay retaining on voltage
This is not common ,
Relay near end lifetime
Bad power supply
Bad signal also means bad contacts, corroded ones.
The buzzing it's something switching on and off.
Sounds like a relay,slow switch not more than few kHertz.