When you say you have really tight calves, whereabouts do they feel tight? I found this when I first started running, especially at the back of my calves. They didn't generally hurt too much while running, but afterwards they would really seize up. I started gently stretching before a run, then following the run with a really good, thorough stretch. This really seemed to help, as did starting off gently at the beginning of a run.
I honestly don't think it's posterior compartment syndrome: it's much more likely that, as you have not been running for very long, your legs just need to get used to the extra use they're getting. Mine went away after a while - hopefully yours will, too.