I had a few bouts of shin splints when I first started running. Here are a few things that helped me get over them quickly:
1. Use an ice cube to massage the inside edge of your shin, where it hurts, and with a lot of pressure, until the pain starts to decrease as the scar tissue is massaged away. Do this daily until you are pain-free
2. Avoid an overly-pronounced toe strike. The closer to level your foot can be when striking the better, and make sure your lower leg is roughly perpendicular with the ground at footstrike
3. Roll out frequently, especially the calves, and stretch regularly as well.
4. Sit in a chair with a shin perpendicular to the ground and tap your toe as hard as you can on the ground. Focus on using the muscle on the front side of the shin, and try to do about 50 reps after you begin to get the muscle tired. Do both sides, 2-3x every other day. Also, try doing this right before you go out and run, but keep the reps to a more comfortable range.
5. Do a small set of glute bridges and core exercises before running- this is more of a generally good habit to make sure your trunk is stable and active, which can help with lower body issues
6. Once you stop running and get the pain to a manageable level, I would say running easy with Advil is okay unless the pain starts to really escalate again– if that happens you should probably take a longer amount of time off.
I know this doesn't really address your question about shoes, but this is what has helped me. I would just make sure the shoes you are using are good running shoes by checking in with a running store. Generally, more structured or cushioned shoes have actually made shin splints worse for me, so I would not recommend going that direction to try and fix your shin issue.