Lets see...
They fought, as you said, in the cartoon. It was fairly evcen, but in the end Asajj managed to gain the upperhand, but Anakin still won because of his anger.
Next time they fight, I believe, is in The Dreadnaughts of Rendili, Part 3 about 6 months before ROTS. Asajj is winning by quite a bit, but Anakin gives in to the Dark Side yet again, winning by a slim margin.
Asajj survives, and , in Obsession, overpowers Obi-Wan (knocking him out of the building) about 4 months before ROTS. She then takes on both Kenobi and Anakin at the same time, Neither is tryng to kill her and each successfully defend themselves against her. Later, when she tried to kill Obi-Wan from behind, Anakin nearly killed her yet again.
Anakin managed to defeat her in The Dreadnaughts of Rendili, Part 3, and she probablt didn't get much stronger between then and Obsession, since she was healing from her wounds the entire time. Anakin, however, grew alot stronger during the six months in-between The Dreadnaughts of Rendili, Part 3 and ROTS, strong enough, in fact, to take out Dooku. He won't need to use the Dark Side to take her out in this instance.