How to find every negative cycle in a graph
First of all, there can be exponentially many negative cycles, as @Steven already mentioned.  Take $K_n$ with only negative edges.  Enumerating all of them will be bound to take exponentially long time.
You also say that you want to find the most negative cycle in the graph (which should be simple if you enumerate them all), however, in case you want a special algorithm for this problem, note that this problem is NP-complete; the reduction from Hamiltonian is straight forward.
Since you ask for any algorithm, just run, for every subset of vertices, a DP algorithm to find the Minimum Hamiltonian Cycle for that induced subgraph.  (Since you don't want to revisit an edge, you can probably find a way to add $\min_{e \in E(G)} w(e)$ to every edge, thereby avoiding all the negativity.)
(P.s., OP stated in a comment that cycles on the same vertex set are considered equivalent.)