Rules:
1. When the 'damping field' is engaged or disengaged momentum is preserved for the spacecraft as observed by all outside observers.
1.1. The spacecraft does not accelerate in response to the field being engaged or disengaged.
2. When the field in engaged for a time, thrusters fired, and the field disengaged, the spacecraft and expended propellant will preserve conservation of momentum, as if the field is not used.
Effect:
a. When propellant fired from a thruster exits the field the field must give the propellant greater momentum to balance the momentum the ship will have when the field disengages.
a.a. This would drain energy from the field, to be replenished by the machinery.
a.b. The field effects a force between propellant exiting the ship and the whole mass of the ship. Thus thrust is generated, but objects within the ship wouldn't feel the resultant thrust.
b. Objects entering the field from outside may be expected to feel the same effect in reverse, and slow relative to the ship. This effects a kind of bumper field reducing the severity of impacts with severe debris.
b.a. This would add energy to the field, which the machinery may have to manage.
c. The net result expends propellant and energy as by a rocket of similar performance.
d. Light entering and leaving an active field would also be effected, incoming light red shifted and outgoing light blue shifted.
Utility:
This type of device would be most useful as an upgrade to a high G torch ship, delivering no substantial improvement in speed but reducing the acceleration felt to sustainable levels. The bumper field effect reduces the impact on whatever other shield system is used.
This field may also be of use augmenting a Buzzard collector which is otherwise excessively lossy.
How it works:
Sci-fi magic.

Have I missed anything?