While running, if you are bouncing up and down vertically too much, that could indicate that you are pushing off, or extending from the ankle, too soon during the stance phase. That would be equivalent to doing calf raises in the weight room for each step.
Like some of the others, I'd say have someone check your form. Good fitting shoes should help, ensure you have enough room in the toe box. And, what you might be doing is crunching up your toes! That'll play havoc with your shins (mine do that in ice skates.)
A rarer, but potentially serious cause of shin pain is Anterior Compartment syndrome of the lower leg. Four muscles of the front of the leg pass through a tight, enclosed, non-expansible compartment. With exercise, there is swelling, and since the compartment won't expand to accodate the swelling muscle, blood circulation decreases and can cause pain at a minimum, necrosis of the muscle tissue at its worse. Like I said, it is rare.
You may also simply be overstepping in your running gait and landing on your heel out in front. The anterior shin group muscle have to contract vigorously and forcefully to prevent the foot from "slapping down". That with coming off a long sedentary stretch of time. I started running after over 20 years of not. It takes time especially for the tendons and ligaments to thicken, hypertrophy and adapt even if them ucles and cardiovascular system are doing well. Be patient and take it slow.
Try a smaller step distance (shorter gait) and allow time to adapt. Good luck, but by all means don't stop the active lifestyle. It has been a lifechanger for me now over 50.