Close Andy, the MK1 LC and the MX's share the same chassis and swing arm though, including the forks etc. The tank/seat and bodywork are different.
You really don't need to mess with a new carb, the VM carbs used on the DT's are pretty idiot resistant. An overhaul kit from motocarb (think thats what its called) will get you running lovely if there is a serious problem with the jets.
Strip out the carb again and run all the jets through with some fine copper wire, kettle flex stripped back is good. The copper doesent damage the brass but is man enough to shift what can be surprisingly stubbon particles of cr@p.
A particular favorite is the idle jet in my experience, although yours is running ok at idle so your ok there. The fact that it dies at a touch of throttle means the main jet needs looking at, and of course the needle height.
I may have missed a mention of it somewhere but you HAVE put a new plug in. 2strokes are incredibly ficke when it comes to the plugs.