We introduce a new class of hybrid Lur'e dynamical systems where a sector nonlinearity may affect both the continuous-time evolution and the reset map acting on suitable closed-loop states, under a time-regularization mechanism ensuring dwell time of solutions. For this class of systems we characterize Lyapunov-based exponential stability conditions exploiting homogeneity of the closed loop. In particular, we show that, with quadratic Lyapunov certificates these conditions can be cast as linear matrix inequalities. We then focus on the control design problem, where both the feedback gains acting on the continuous evolution and the reset action must be designed, in addition to the sets where such resets are triggered, expressed by sign-indefinite quadratic forms. For this control design problem we also show that the synthesis can be performed by solving a set of linear matrix inequalities.