Adaptive feedback linearizing control schemes are used to suppress limit cycle oscillations in nonlinear systems where the system parameters are either unknown or uncertain. Parameter convergence is desirable in these schemes as it provides a measure of robustness of the scheme and also permits the unknown/uncertain system parameters to be estimated. In recent work, we have shown how using a persistently exciting forcing it is possible to achieve parameter convergence in nonlinear limit cycling systems. In practice, however, limits on the control input to the plant due to saturation must be considered, and the main goal of this work is to analyze the effect of input saturation on parameter convergence in an adaptive feedback linearization framework. In particular, a technique known as control hedging is incorporated and the effectiveness of this method for very severe saturation constraints has been evaluated. Results are presented for a single degree-of-freedom wing rock dynamics model and a multi degree-of-freedom combustion acoustics model showing successful parameter convergence even in the presence of input saturation.