We report on a new implementation and first numerical tests of the fourth-order algebraic-diagramatic construction [ADC(4)] propagator method for K-shell ionization in molecules. The theory, which has been presented in a preceding paper (paper I) [J. Chem. Phys. 115, 10621 (2001)], is based on an intermediate state representation (ISR) or non-Dyson reformulation of the general ADC approach and uses the core-valence separation approximation to specialize the method to the case of core-level ionization. The ISR form offers the possibility to go beyond the finite perturbation-theoretical expressions of the previous (strict) ADC(4) scheme, and several such modifications are considered. The general aim of the present development is to establish a practical "frozen" orbital method that can be applied equally well to systems with and without equivalent core levels. A set of small model calculations has been carried out on the 1s ionization in CO and N-2 allowing us to compare the ADC results with those of large-scale configuration interaction treatments and analyze the performance of several ADC(4) modifications in describing the 1s ionization energies as a function of the bond lengths. These studies clearly demonstrate the need for improving the previous (strict) ADC(4) scheme. In the theoretical description of the C1s and N1s ionization, the proposed modifications lead to a substantial improvement, whereas in the case of the O1s ionization further developments appear to be necessary. In another set of calculations using reasonably large basis sets, the improved ADC(4) method is applied to the vibrational structure in the CO C1s and N-2 N1s ionization spectra. The results are in very good agreement with the findings of recent high-resolution measurements. An interesting feature is the different behavior of the g and u components of the N1s ionization energy curves of N-2 leading to different vibrational line intensity distributions. (C) 2003 American Institute of Physics.