Coring tools configured to procure core samples of earth formations may include a coring bit comprising a cutting structure configured to cut a core sample and an outer barrel connected to the coring bit. The outer barrel may be configured to apply axial and rotational force to the coring bit. An inner barrel may be located within the outer barrel and may be configured to receive a core sample within the inner barrel. A sponge material may line an inner surface of the inner barrel and may be configured to absorb a fluid from the core sample. A stabilizer may be connected to the outer barrel. At least one blade of the stabilizer may be rotatable with respect to the outer barrel and may be configured to remain at least substantially rotationally stationary relative to the earth formation during coring.