In a non-symmetric turn, how do the side slip angle and thrust side slip angle affect the equations of motion?
Relying on physics, causing a mass to execute a circular path requires a suitable centripetal force. In an airplane in a banked balanced turn, this comes from the inward component of lift due to the bank angle. 

However, if the turn is not properly co-ordinated, there can be additional contributions to the centripetal force from the power plant thrust no longer being directed tangentially and sidethrust on the airframe due to slip.

The extreme case is the flat turn where the wings remain level and the power plant thrust has an inward component due to the sideslip.

That said, apart from an airplane having vectoring thrust, the only reason for a thrust sideslip angle I can think of is that many single airscrew airplanes have built-in sidethrust to counteract engine torque. However there is no requirement for the engine sidethrust angle to equal the slip angle, so in that case the assumption that the two angles are the same cannot be valid.

The thrust line of an airscrew doesn't change with slip. The slip causes asymmetric thrust on the blades which precesses gyroscopically so the airscrew creates a couple which tends to increase the slip angle.