We study the Drude weight D and the total optical weight K for a t - t' - J model on a square lattice that exhibits a metallic phase-modulated antiferromagnetic ground state close to half-filling. Within a suitable 1/N expansion that includes leading quantum-fluctuation effects, D and K are found to increase linearly with small hole doping away from the Mott metal - insulator transition point at half-filling. The slow zero-sound velocity near the latter transition identifies with the velocity of the lower-energy branch of the twofold excitation spectrum. At higher doping values, D and K eventually saturate and then start to decrease. These features are in qualitative agreement with optical conductivity measurements in doped antiferromagnets.