I would recommend not running for a good 2 weeks to allow full recovery. Pushing through a run when you are experiencing pain may lead to a change in your running form which could produce more issues up/down your kinetic chain. Also running through the pain could lead to a more serious condition--a stress fracture that could leave you side-lined for more than a few weeks (8-12 weeks).
As for shoes, they may be the cause, but it also could be the fact that your anterior tibialis muscles are not conditioned to running. It could also be the type of running surface you are running on--concrete being a big culprit.
Just ice your shins up and do these exercises...they may help strengthen the tendons and ligaments in the lower leg.
Sit in a chair with your leg extended in front of you. Trace the alphabet in cursive with your toes trying not to break the rhythm. Do this on each leg 3-5 times, 2-3 times a day.
Put marbles on the floor and pick them up with your toes and place in a cup. Do this 1-2 times daily with each foot.
Place a towel flat on a wood/tile floor. Sit in a chair and then use your toes to scrunch the towel in a heap. Do once on each foot daily.
Fill a tube sock with gravel or coins (that you find on your walk/runs....LOL), seal with rubber band. Sit down and drape the sock with the coins or gravel equally distributed over your foot....then lift your foot toward the ceiling. Do this 10 times each foot several times a day.
You do not have to do all these exercises, I just gave several options, but you will want to do these to help strengthen the lower leg.
I hope this helps!