BEAM188 is suitable for analyzing slender to moderately stubby/thick beam structures.
Yes, it is true that BEAM188 is based on Timoshenko beam theory which includes shear-deformation effects. The element provides options for unrestrained warping and restrained warping of cross-sections.
The transverse-shear distribution calculation ignores the effects of Poisson's ratio. The Poisson's ratio affects the shear-correction factor and shear-stress distribution slightly, and this effect is ignored.
Shear-correction factors for some common sections are as follows: rectangle (5/6), solid circle (9/10), thin-walled hollow circle (1/2). These are fixed values and cannot be changed.
For the arbitrary section (ASEC), shear-correction factors can be input directly using the command SECDATA.