We calculate 2D electronic-vibrational (2D-EV) spectra of solvated organic dyes modeled in terms of a reduced set of electronic diabatic states (the essential states) non-adiabatically coupled to molecular vibrations. An effective overdamped coordinate, whose dynamics is described by the Smoluchowski diffusion equation, accounts for polar solvation. Results are discussed for two dyes with distinctively different spectroscopic behavior: 4-(dicyanomethylene)-2-methyl-6-(4-dimethylaminostyryl)-4H-pyran (DCM) and 8-(N,N-dibutylamino)-2-azachrysene (AAC). Linear absorption and fluorescence spectra of DCM are well reproduced based on a minimal two-state model. The same model leads to 2D-EV spectra in good agreement with the recent experimental data reported by Oliver and coworkers for DCM in DMSO. In contrast, linear spectra of AAC show a subtle interplay between a locally-excited (LE) and a charge-transfer (CT) excitation, calling for a three-state model. Calculated 2D-EV spectra for AAC show a qualitatively different behavior, demonstrating that the experimental data for DCM do not support a LE/CT interplay. This resolves the long-lasting discussion about the nature of low-lying excitations of DCM in favor of the simplest picture.