Applications of the Feshbach formalism to systems of more than two active electrons are very scarce due to practical limitations in the construction of the projection operators $\mathcal{P}$ and $\mathcal{Q}$ that are inherent to the theory. As a consequence, most previous applications rely on the use of approximate quasiprojection operators, whose theoretical justification is not yet clear. In this work, an implementation of the Feshbach formalism for three-electron atoms is presented that includes all the ingredients of the original formalism. Energy positions and autoionization widths of the lowest ${}^{2}{S}^{e}$, ${}^{2}{P}^{o}$, and ${}^{2}{D}^{e}$ autoionizing states of Li and Ne${}^{7+}$ have been evaluated. The results show that the use of quasiprojection operators is justified for the evaluation of resonant positions. However, for the ${}^{2}{S}^{e}$ states of Li, the use of quasiprojection operators can lead to errors in the autoionization widths of the order of $100%$.