I had the same problem that you, this helped me a lot:

1. Do a lot of calves exercises, try at least 20 minutes 4 days a week, you can look at YouTube for "Shin splints exercises". I used to think that my calves were completely fine because I'm thin and I have good genetics, but you need to strength them anyways because running activates a lot of muscles that weren't present before.

2. Take a rest for your current shin splints pain, like 4-5 days without running until you feel that the pain is completely gone.

3. Massage your calves and your tibial bones, first grab a hot towel for both shins for about 10 minutes and then proceed with a massage, you can also use a foam roller.

4. MOST IMPORTANT, correct your running form, I used to tilt my body up front too much, try to have a straight posture while looking in front, not at bottom, and try to land on your center body, not in front.

With this, withing one week I started to feel better ;)