*"Morse-Bott approach to contact homology"* (thesis of Bourgeois) shows you how to handle the degenerate scenario (and Chris Wendl's thesis will explain more of the Fredholm theory). Everything's fine because for generic contact forms we have nondegeneracy, so you can always perturb things. In particular, we can take a sequence of nondegenerate contact forms $e^{f_k}\lambda$ converging to the degenerate one.

With nondegeneracy, the orbits are isolated and that helps greatly with computations (and constructions where you modify structures on neighborhoods of the orbits). In particular, for computing Conley-Zehnder indices.