A strong version of the loop theorem implies that if an essential closed curve on the boundary of a 3-manifold $M$ is nullhomotopic, then realizing its image in $\partial M$ as a 4-valent graph, you can draw a cycle with no edge repeats that is homotopic on $\partial M$ to the boundary of an essential embedded disk in $M$. (Compare with Bing's "The geometric topology of 3-manifolds", pg 205.)

Q: Is there a similarly strong version of the annulus theorem?

For instance, suppose you have a (say, hyperbolizable) 3-manifold M with incompressible boundary, and you have two closed curves on the boundary of M that bound an essential singular annulus in M. (E.g., they're homotopic in M but not within its boundary.) Homotoping them to be in general position, can you find two cycles with no edge repeats on the 4-valent graph that is the union of their images, such that the two corresponding curves are homotopic to the two boundary components of an essential embedded annulus in M?

The closest thing I've found is Thm 2 in Cannon-Feustel's "Essential embeddings of annuli...", but that requires the two original curves to be disjoint.