Cascaded quadratic nonlinearities from phase-mismatched second-harmonic generation build the foundation for robust soliton modelocking in straight-cavity laser configurations by providing a tunable and self-defocusing nonlinearity. The frequency dependence of the loss-related part of the corresponding nonlinear response function causes a power-dependent self-frequency shift (SFS). In this paper, we develop a simple analytical model for the SFS-induced changes on the carrier-envelope offset frequency (fCEO) and experimentally investigate the static and dynamic fCEO dependence on pump power. We find good agreement with the measured dependence of fCEO on laser output power, showing a broad fCEO tuning capability from zero up to the pulse repetition rate. Moreover, we stabilize the relative intensity noise to the -157 dBc/Hz level leading to a tenfold reduction in fCEO-linewidth.