In a previous work we derived an equation for the radial distribution function g(r) for molecular liquids. It accounted for density correlations at both the monomer and molecular level. Here, it is shown that the theory can be simplified to a form than allows it to be solved easily by standard numerical methods. The theory is applied to charged, rodlike polymers with explicit counterions in solution near the idealized counterion condensation threshold (lambda(B)/bsimilar to1, where lambda(B) and b are the Bjerrum and chain bond length, respectively). For densities above chain overlap, rho(*), it is found that the counterion cloud is diffuse about the polymer with a range on the order of the Debye-Huckel screening length. It is shown that the scaling with density of the first nonzero wave vector peak k(max) of the polymer-polymer partial structure factor agrees with experiment and previous theory, with k(max)similar torho(nu) and nuapproximate to1/2 and 1/3, for densities above and below rho(*), respectively. It is also found that the ratio of the full width at half maximum of the peak, Deltak, to k(max) is a minimum near rho(*). On the other hand, for the counterion-counterion partial structure factor it is difficult to find any sharp scaling of k(max) though the apparent exponent for the semidilute and a large part of the dilute region is roughly approximated by nu=2/5. Asymmetric solutions are also discussed.