Hi Spam, have you looked at the Shin-splints & Tibial periositis sections in the injuries section of this website? There is some quite useful info there (although both sections seem to describe exactly the same injury). Many runners (including myself) get these problems, especially when new to running, or increasing the amount of running. Although you have to be careful that you don't have/get stress fractures, you may well find that by running off-road, and icing the area post-runs, that you can run through it. Stability running shoes are a must too, as it is excessive pronation that causes overworking of the tibialis posterior muscle that is often the problem.
If it doesn't improve, do see an expert!
Good luck.