Twin T circuit noisy
The "buzz" is the usual mains hum, picked up by the body as an antenna and transferred when you touch it. Breadboards tend to be very good at picking up ambient noise anyway, and capacitatively coupling signals that should not be coupled. 

Try running the whole thing off a battery; that will tell you whether PSU noise is a problem. (Are there two power supplies or does the 9V wall supply appear twice?)

The resistors in the filter seem very large, which intuitively suggests poor noise rejection around the negative input of the opamp. That's the location I'd focus on for noise avoidance. Tidy up the breadboard in that area - keep wires short.

The long "pulse" wire might also be a noise antenna, and might benefit from a high value resistor to ground.