We investigate in detail the phase diagrams of the p-body ±J Ising model with and without random fields on random graphs with fixed connectivity. One of our most interesting findings is that a thermodynamic spin-glass phase is present in the three-body purely ferromagnetic model in random fields, unlike for the canonical two-body interaction random-field Ising model. We also discuss the location of the phase boundary between the paramagnetic and spin-glass phases that does not depend on the change of the ferromagnetic bias. The independence of the ferromagnetic bias has also been observed from the perspective of the distribution of the partition function zeros on the complex field plane. This behavior is explained by a gauge transformation, which shows that gauge-invariant properties generically do not depend on the strength of the ferromagnetic bias for the ±J Ising model on regular random graphs.