Consider having this medically evaluated.
If you indeed have shin splints, you have no choice but to avoid running totally!
Every time you go out running, no matter what shoes you wear or precautions you take, you are at great risk of re-injuring your shins.
It's better to get bored on your bike, than developing complications from altered running mechanics which can cause chronic issues and affect your ability to run again.
It might take a few months to heal.
I had a similar issue when I played University soccer, which was due to frequent starts and stops during training and the fact that I'm flat-footed (risk factor).
This caused me to miss half the season as it took a few
months to heal properly.
Best wishes,
Niko