I am type II diabetic and run daily up to 20K, congrats at getting out there and exercising. More people should be exercising diabetic or not. So do not get discouraged by the fact that your blood sugar spikes up during exercise.
First, are you checking your blood sugar before bed? If you are waking up with a BS over 6 chances are you are going to bed with higher then normal BS. So first I would work on getting your BS into the 4.5 to 5.5 range before bed. BS does not change much between the time you go to bed and wake up.
Your liver will dump in the morning if it perceives you need sugar. So try a ½ cup of low fat yogurt or even a Rice cake with peanut butter on it before heading out the door. I find that once I get moving my BS will start to be used very quickly in the morning so there tends to be a faster drop in the mornings then other times of the day.
Also I do not skip out the door to run I generally give myself a ½ hour to 1 hour before setting out. This allows for the body to wake up and start functionally normally. So I might suggest giving your self a bit of time before exercising in the morning just to let the body ramp up into the normal functions.
This may sound funny but you could be stressing the body into the Fight or Flight mode. If your body is being stressed the liver will dump sugars into the blood stream. So by getting up strapping on the running shoes and heading out the door, your body may tricked into thinking there is a need for an extra boost of energy. So give yourself that ½ hour to read the paper have a small meal. Then go out and walk.
Harry