We present quasiparticle (QP) energies from fully self-consistent $GW$ (sc$GW$) calculations for a set of prototypical semiconductors and insulators within the framework of the projector-augmented wave methodology. To obtain converged results, both finite basis-set corrections and $k$-point corrections are included, and a simple procedure is suggested to deal with the singularity of the Coulomb kernel in the long-wavelength limit, the so called head correction. It is shown that the inclusion of the head corrections in the sc$GW$ calculations is critical to obtain accurate QP energies with a reasonable $k$-point set. We first validate our implementation by presenting detailed results for the selected case of diamond, and then we discuss the converged QP energies, in particular the band gaps, for the entire set of gapped compounds and compare them to single-shot $G_0W_0$, QP self-consistent $GW$, and previously available sc$GW$ results as well as experimental results.