In this work, we consider the problem of estimating the parameters of polynomially damped sinusoidal signals, commonly encountered in, for instance, spectroscopy. Generally, finding the parameter values of such signals constitutes a high-dimensional problem, often further complicated by not knowing the number of signal components or their specific signal structures. In order to alleviate the computational burden, we herein propose a mismatched estimation procedure using simplified, approximate signal models. Despite the approximation, we show that such a procedure is expected to yield predictable results, allowing for statistically and computationally efficient estimates of the signal parameters.