Shin Splints are an indication that something is not right in your biomechanics. You may need more support or more cushioning. Some people get good results from orthotics, others get good results from very cushioned flexible shoes.
I would speak to a knowledgeable physio-therapist or podiatrist. They can assess your mechanics and get you on the right path. 15 minutes with a PT is more efficient than several weeks and hundreds of dollars wasted on the wrong shoes.
Also "shin splints" can also be a stress fracture of the lower leg. No shoe is going to correct for damage to your bone. A PT will assess for that as well.
In the mean time, ice your shins after running and at night. Flexibility exercise can help. Do a goggle search.