When solving differential delay equations (DDEs) with state- or time-dependent delays, the presence of vanishing or near vanishing delays can make it impossible or impractical to use step-sizes which are smaller than the delays. The phenomenon of the delay being smaller than the step-size is referred to as "overlapping". When overlapping occurs delays fall in the current computational step, and the corresponding solution values are computed by interpolating between the stage values. In this case even explicit methods become fully implicit. Runge-Kutta methods which remain explicit even in the case of overlapping were first introduced by Tavernini for Volterra functional differential equations, and more recently developed as functional continuous Runge-Kutta methods by Maset and co-authors. Diagonally implicit Runge-Kutta methods can have better stability than explicit methods, but were not previously explored for state-dependent DDEs because standard methods become fully implicit in the case of overlapping. In the current work we apply and extend the techniques of the Italian school to create diagonally implicit Runge-Kutta methods which retain their diagonally implicit character even in the case of overlapping. Various methods and order bounds will be presented, including fourth order methods.