First of all, GOOD FOR YOU
It's great that you've given running a try and you've come a long way!
When I first began running, I had a sharp pain in the muscle in the lower part of my right calf (sort of halfway between my actual calf muscle and the achilles tendon). It was quite intense, and I had to stop and walk several times during many runs. Unfortunately, the pain didn't seem to ease with walking either.
help, however, was taking a few easy running days. I took a few days where I'd go out for just a mile or so. I even took a day or two off from running altogether. After that, the pain went away. I read a fair amount about it back then, and it seemed likely that it was a very mild injury due to increasing my activity level all of a sudden, and then doing too much too fast. I don't know where you started out, but it's possible if you started the way that I did, given a few days to rest, your calves will be back in order.
Either way, GOOD LUCK
to you in your 5K! You'll do great, and the feeling of accomplishment will stay with you for days to come!