This paper reports a new design of a tunable optoelectronic full-adder using two photonic crystal ring resonators (PCRRs). Every PCRR consists of a matrix of silicon rods surrounded by silica rods covered with graphene nanoshells (GNSs). The proposed full-adder is formed by three input ports, two PCRRs, and two output ports for 'SUM' and 'CARRY'. The plane wave expansion technique is used to study the photonic band structure of the fundamental PC microstructure, and the finite-difference time-domain method is also employed in the final design for solving Maxwell's equations to analyze the light propagation inside the structure. We can tune the PhC resonant mode for our desired application by setting the chemical potential of GNSs with an appropriate gate voltage. The numerical results reveal that when the chemical potential of GNSs changes, the switching mechanism occurs and manages the coupling and propagation direction of the input beam inside the structure. We systematically study the effects of physical parameters on the transmission, reflection, and absorption spectra. Our numerical results also demonstrate that the maximum delay is about 0.8 ps. The 663 µm2 area of the proposed full-adder based on two-dimensional materials makes it a building block of every photonic integrated circuit used for data processing systems.