You do not get any spirals. In Newtonian gravity two-body motion is always along conic sections.
The "trick" here is to look at the problem from a moving reference frame. For simplicity, assume A and B to have equal mass. Then if you move with the centre of mass for the whole system you will see A and B start out with equal and opposite velocities and perform the usual elliptic, parabolic or hyperbolic orbits. With a bit more algebra one can turn the 2-body problem with unequal masses into a 1-body problem, and again the orbits are the standard conic ones. When you transform back to the original view you can certainly see A and B whirling around each other while the pair is also moving in some direction. But you never get any inspirals.
Three-body interactions are way messier, and I think there are at least mathematical cases where bodies collide after an infinite number of orbits around each other. Things are also slightly different in general relativity, where you can actually get spiral plunges close to black holes. In any of these cases I don't think the spirals are neatly described by any of the standard spiral formulas.