Yes, you are correct. Both beam elements BEAM188 & BEAM189 elements are based on Timoshenko beam theory which includes shear-deformation effects. The elements can account for unrestrained warping and restrained warping of cross-sections.
The amount of error generally depends on approximations we make in our model. Can you check with both beam elements and check what’s the difference in error. The amount of error you are having is acceptable and doesn’t need to be worried about.
For more information on beam elements, kindly refer to below links: