  Mechanical resonators based on low-dimensional materials are promising for
force and mass sensing experiments. The force sensitivity in these ultra-light
resonators is often limited by the imprecision in the measurement of the
vibrations, the fluctuations of the mechanical resonant frequency, and the
heating induced by the measurement. Here, we strongly couple multilayer
graphene resonators to superconducting cavities in order to achieve a
displacement sensitivity of $1.3$ fm Hz$^{-1/2}$. This coupling also allows us
to damp the resonator to an average phonon occupation of $7.2$. Our best force
sensitivity, $390$ zN Hz$^{-1/2}$ with a bandwidth of $200$ Hz, is achieved by
balancing measurement imprecision, optomechanical damping, and heating. Our
results hold promise for studying the quantum capacitance of graphene, its
magnetization, and the electron and nuclear spins of molecules adsorbed on its
surface.
