Despite the molecular strong-field approximation (SFA) theory has made remarkable achievements in describing the ultrafast dynamics of molecules in intense laser fields, there are basic inconsistencies in the theory itself. On the one hand, the basic principle of SFA requires that the initial state be an eigenstate of the system in the absence of the field, and on the other hand, the spatial translation invariance of the physical process requires that the initial state of the system be a laser-field-dressed state. These two conflicting requirements correspond to the two forms of molecular SFA theories, namely, the undressed state and the dressed state. The two theoretical validity and applicability conditions are widely disputed. In this paper, we investigate the ionization processes of N₂ and Ne₂ molecules in an elliptically polarized laser field and a circularly polarized laser field, aiming to solve the above-mentioned controversies. Elliptically polarized laser can efficiently suppress the re-scattering process and the influence of various interference effects, which makes the ionization process cleaner, and thus can effectively screen the applicable conditions for the dressed and undressed states. We calculate the photoelectron momentum distributions corresponding to different molecular orbitals in the dressed and undressed states by using the SFA and the Coulomb-corrected strong-field approximation and compare them with previous experimental results. For molecules with large nuclear spacing such as Ne₂, we find that the dressed state is necessary to accurately characterise their ionization, however, for molecules with small nuclear spacing such as N₂, the dressed state description is inapplicable. The conclusions of this work provide a reference for accurately describing laser-induced molecular ultrafast processes and further developing corresponding theories and molecular ultrafast imaging schemes.