We investigate the eccentricity effects and depth of investigation (DOI) of two typical types of azimuthal resistivity Logging-While-Drilling (ARLWD) tools (Type I: an axial transmitter with a transverse receiver; Type II: an axial transmitter with a tilted receiver). A three-dimensional alternative direction implicit finite difference method in time domain (ADI-FDTD) is used to numerically simulate the tools' responses with different eccentering offsets and azimuth angles in the complex borehole condition. To improve the accuracy and speed of the simulations, we adopt several strategies in the ADI-FDTD algorithm: (1) a 3D cylindrical grid and non-uniform meshing method; (2) a depth window to limit the computational model; (3) locally-conformal FDTD (LC-FDTD) technique to realize fine mesh of an eccentering tool. Based on the numerical simulations, it is observed that even for a centralized ARLWD tool, the geosignal acquired in a conductive mud filled borehole (less than 0.1 Ω m) with a short transmitter-receiver spacing (16 in) and high operation frequency (2 MHz) is distorted, which indicates the ARLWD tools prefer an oil-based drilling mud borehole. The geosignal of both types of tools changes linearly with eccentering offset. Furthermore, the geosignal shows a trend of cosine curve with different eccentering azimuth (the azimuth is zero when the tool moves perpendicular to the formation boundary). The DOI of Type II tool is affected by both transmitter-receiver spacing and operation frequency. However, the DOI of Type I tool is mainly determined by the frequency, which makes the ultra-deep detection with a short spacing possible.