Using the coherent potential approximation (CPA), we study the absorption spectra of two-dimensional molecular aggregates formed from binary random molecular mixtures. In addition to the substitutional randomness, we include Gaussian randomness in the transition frequencies within each of the two classes of molecules. The latter is motivated by the considerable disorder that is typical for two-dimensional aggregates. By comparing to exact diagonalization results for small clusters, we show that the CPA gives an excellent description of the spectra for this kind of disorder, both in the cases of amalgamation and persistence type mixing. Taking into account long-range excitation transfer interactions mediated by the extended molecular transition dipoles, we analyze experimental spectra of amalgamation-type mixed cyanine aggregates adsorbed on AgBr {100} surfaces that are also reported here. We find good agreement between theory and experiment for the position, the width, and the general shape of the absorption line as a function of the mixing ratio. The analysis also allows us to estimate the slip angle characterizing the structure of these aggregates.