Perhaps I have had too much coffee and Tylenol 3 this morning(found out last night my ankle which I have thought was slightly sprained for past 3 weeks is actually fractured)but I have several different, purely speculative scenarios.
1. EA/CM leave together in his car, he starts driving somewhere which raises her suspicion like the woods, when she gets a chance, struggles with him, jumps out and runs, he hits her with the car, then to try and disguise the damage on his car he does punch/kick it which could explain hands/limp.
2. Basically same as scenario one, except they struggle in car, which causes him to hit something while driving, which further ignites his anger and he kills.
3. He convinces CM he is too wasted to drive, so she drives his car, he gives her directions that lead them to somewhere they won't be seen. They struggle in car, perhaps he gets out to come pull her out of driver seat(I had a car like this for a bit, bucket seats make it really hard to move around in the front and I think it would be nearly impossibly and illogical to pull someone anywhere on that car as there isn't a lot of room), she hits him with car, he gets up obviously enraged and they further struggle until he kills her.
Sorry if these seem loopy, but, in my painkiller induced state right now they all seem logical!
Eta: in number 2 and 3 he would also hit and kick his car to disguise a dent from hitting a person.