Far-from-equilibrium dynamics that lead to self-organization are highly relevant to complex dynamical systems not only in physics but also in life, earth, and social sciences. However, it is challenging to find systems with sufficiently controllable parameters that allow quantitatively modeling of emergent properties. Here, we study a nonequilibrium phase transition and observe signatures of self-organized criticality in a dilute thermal vapor of atoms optically excited to strongly interacting Rydberg states. Electromagnetically induced transparency provides excellent control over the population dynamics and enables high-resolution probing of the driven-dissipative dynamics, which also exhibits phase bistability. Increased sensitivity compared to previous work allows us to reconstruct the complete phase diagram, including in the vicinity of the critical point. We observe that interaction-induced energy shifts and enhanced decay only occur in one of the phases above a critical Rydberg population. This case limits the application of generic mean-field models; however, a modified, threshold-dependent approach is in qualitative agreement with experimental data. Near threshold, we observe self-organized dynamics in the form of population jumps that return the density to a critical value.