We study small rod-like molecular electrolytes solutions with their corresponding atomic counterions.
The asymptotic length scales (decay length and wavelength) of the structural correlations are
analyzed using the formalism of the dressed interaction site theory (DIST). The correlation functions are
determined using the reference interaction site model equation complemented with a mixed approach in
which the hypernetted-chain closure is used for the repulsive interactions, and the mean spherical approximation
is used for the attractive interactions. The results from this scheme are in good agreement with the
Monte Carlo computer simulations reported here. The asymptotic properties of the correlation functions of
this molecular system are compared against those corresponding to two related simple (atomic) electrolyte
models. The main conclusion is that the molecular structure of the ions lowers by two orders of magnitude
the concentration at which the transition from monotonic