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Hi, Dave. I've used the ASICS Nimbus running shoes and they are okay for longer than 15K distances. Sounds to me like your problem is running form and not running shoe. Your shins aren't strong enough for longer distances.
Drop the distance down, and be sure to stretch your calves before and after running. Roll your shins out with a foam roller or marathon stick. And hire someone to teach you proper running technique and form. You are probably over striding and running incorrectly.
Finally, run on grass and trail only. No asphalt! Then, increase mileage by not more than 1/2 mile per week. If you have pain in your shins, ice them.