Exact statistical mechanical sum rules are combined with thermodynamic scaling arguments to determine the leading-order singular contributions to the transverse moments of the density-density correlation functionGof a simple fluid undergoing a complete wetting phase transition, from off-bulk coexistence, at a structureless substrate (wall). Contrary to our earlier suggestion, capillary-wave-like fluctuations do manifest themselves throughout the wetting film and lead to a transverse correlation length ξ‖(a), for both particles near the wall, which diverges with the same critical exponent as ξ‖, the correlation length appropriate for both particles in the depinning liquid-gas interface. Explicit results from a mean-field (density functional) calculation are consistent with our general predictions. In the weak fluctuation regime (d< 3), scaling methods are used to derive the short distance expansions of the density profile and moments ofGfor short ranged forces. These expansions, which describe singular contributions for distances ≪¯l, the average film thickness, are considerably richer than those for critical wetting at bulk coexistence. Using sum rule arguments and the results of a density functional calculation for the case of complete drying at a hard wall, we identify specific corrections to the leading order singular behaviour of the moments ofG. We show these corrections are not accounted for by the simplest capillary-wave-based ansatz forG. More significantly, they are not accounted for by the standard interfacial Hamiltonian description of complete wetting. We show, using sum rule arguments for interfacial Hamiltonians, that singular correction terms can be incorporated if the interfacial stiffness Σ (T) in the standard Hamiltonian is allowed to depend on the (fluctuating) film thicknessl(R). For short range forces we require the increment ΔΣ(l) to be proportional to (T-Tw)le-l/ξb, whereTwis the temperature of the critical wetting transition. This result is the same as the (leading order) result for ΔΣ(l) derived recently using an integral criterion to definel. We discuss other consequences forGof modifying the standard interfacial Hamiltonian.