Sagnac interferometry (Mach-Zehnder type) has been used for the self-Kerr nonlinearity (SKN) dependent photon drag. The presence of SKN results in an improvement in the light drag angles from±10 micro-radians to±1 centi-radians. Additionally, the system exhibits Electromagnetically induced transparency (EIT) with the presence of SKN. Optical angular momentum (OAM) dependency in the probe field is introduced for controlled light dragging by allowing it to pass through a spiral phase plate. Improved positive and negative group indices are noticed respectively, in the presence and absence of SKN. Consequently, reduced relativistic group velocities for the propagating beams are observed when SKN is present in the system. The outcomes displayed has the potential to be used in many different sensing schemes.