HC06 Weak Transmission
The "finger trick" looks to me like a symptom of an unconnected pin. Unconnected pins in MOS circuits can be very high-impedance, and the resistance (and capacitance) of a finger can affect operation. I'd check very carefully for pins/connections that have become disconnected. The break in the circuit could be a breadboard pin, a connector pin, or even a cracked solder joint on the Bluetooth module itself. If the module has suffered a mechanical impact, also check for completely missing surface-mount components.