Shin splints can be caused by several things, but one common cause is tightness or weakness in the muscles on the front side of your lower leg (along your shin). Try some daily stretching and basic strengthening these muscles for the next 2-3 weeks and see if there is any improvement. I would also taper down your running volume during this time to allow some improvements to occur.
Stretching these muscles is simple: remove your shoes then kneel with your toes pointing away from your body. Slowly lean backward until your feel the stretch in the shin area. Hold this stretch for 30 seconds, relax and repeat the stretch 2-4 times. Perform this stretch 1-2 times a day for the next 2-3 weeks. I would also suggest performing this stretch before each run.
To strengthen these muscles anchor some elastic tubing to a fixed object and sit on the floor, facing the tubing with your legs extended. Wrap the tubing around your foot and once secure, slowly perform some repetitions, flexing your ankle, pulling your toes towards your body. Perform 1-2 sets of 8-12 repetitions per foot, alternating the exercise between your feet to allow for muscle recovery.
Other techniques to relieve the discomfort of shin splints include cold down stretching of these muscle and 10-15 minutes of icing the shin areas after your run and post-exercise stretching. If these techniques don’t result in any improvement, I would encourage you to speak to your doctor or a podiatrist (foot specialist) who could complete an evaluation of your running shoes and running mechanics and make some professional recommendations.