In theory, this is easy to do, just wake up early and use the shower in the gym. The problem is that using the students' locker room for changing and showering is strongly frowned upon for faculty. Some schools will have no problem, others will strongly frown upon it. Check with an administrator, less you get accused of being a pedo. You also run into issues with running back home, what if its real hot or storming, or the roads have been covered with snow, but not plowed.
Next, what if you have to take home stuff to grade? You're not going to run home with a big stack of 150 papers. Lets say you have two, one page assignments you collect, thats 300 pages to take home that night. I suppose you could drive back to school and grab them after running home, but that defeats the purpose. Sometimes you'll be having 600+ pages to lug around. Try as hard as you like, you won't get through all that in your planning periods.
How much time do you really save over starting and stopping your commute at home, showering there, and then driving to school. Only the time of your drive. I could see it being worth it if say school is over an hour commute away due to traffic, but not any further by run, but then you live in an urban area and probably have better options.
As a young teacher, you do not want to buck the trend. Even if all the above seems positive in terms of running to work (supportive administration, ease of commute, a way of getting stuff and supplies to and from, ability to shower at work) you're still the weird guy who runs to school. Bobby and fattty Sally run too. Fatty sally ran 3 half marathons last year and she even broke 3 hours. You haven't run further than 10k right? (rhetorical, and likely technically true if you only count race distances and you're right out of college). If your commute is under half an hour, I just think its more worth it to get up earlier and finish runs at home, then drive to school.