It sounds like your asics shoe will do except for traction in the mud. If you like the asics, then you can look for one of the Asics trail runners or the cross over. I have had good luck running the Fuji, the Trabuco, and the Kahana. I have since moved to running Hokas. For me it is more about the stability that the certain models offer because I over pronate A LOT!! As for the others mentioned... I like the Salomans for known muddy terrain. When just wearing them, I found that I tore and wore out some of the lugs thus causing more movement of the foot/shoe on the trail. I also run a lot over rocky terrain and granite surfaces so lugs are out for me unless the trail is muddy and slick. I have tried the altras as well. I like them as an everyday shoes, but they don't work for me as runners with the zero drop.
The best advice is to start with something close to your road shoe but maybe just a tad more traction on the sole. A good runners shop can watch your movement when running and recommend something better.
Nonetheless, enjoy the run!