Yes they do, but I don't think people ever worry about actually writing them out in nuclear physics. Protons have a wavefunction comprised of a spatial part and a spin part (and also an isospin part if you treat them as indistinguishable). For some reason nuclear physicists also assume that the neutrons and protons are bound in a more or less Coulomb potential, so that they can use the shell model to describe the nucleus. But when I took nuclear physics, we only used the nuclear wave functions at an abstract level to determine if nucleons were in a singlet or triplet state, spin-orbit coupling perturbations, etc.