Shin splints happen because your feet are not shaped in such a way that they distrubute the impact of running correctly. A trampoline would be fine, because its not putting jarring impact on your shins and therefore they won't get sore.
I had the exact same problem being on the tennis team in my high school...I could barely play because it hurt so much for me to run. But then I went to an orthapedic doctor, he got me hooked up with some 150$ a pair shoe insoles (they are custom molded to my foot and make my arch fall correctly, and theyre burly as hell, each pair lasts 1-1.5 years and I jog and am active) and that COMPLETELy solved my problem after a month.
I highly suggest you do see an orthapedist. Once you get insoles and your feet get used to them after a couple weeks of use, you should never get shin splints again.
(note the insoles at shoe stores to DO NOT work for this...they only pad the shoe. the ones from the orthapedist are a special compound that when warmed for the first time will mold to fit your arch, and then they stay that way...theyre kind of hard and uncomfortable at first until your foot conforms to but it but from then on you're golden.)
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