Advice please – Ongoing calf pain

I have been training for 15 weeks and am now running around 5m three times a week, and a Sunday run of around 8m. I never push myself very hard on a run, and am always able to breath and speak easily. I make sure I am properly hydrated.
The problem I am having is painful calf muscles. Lately they always seem to be hurting, and also my achilles heel. It is not an aching pain, more of a sharp pain in a specific part of the calf – (normally the middle to top area).
The pain usually subsides after a few days rest, then when resuming running the other calf will start to hurt. My quads are very strong and never ache after a long run, and I always take time to lightly stretch 5min into a run and then longer stretches afterwards.
I’m pretty sure I am a neutral runner. My current shoes are Asics Gel which my running shop tells me are ideal for me – they did not try to sell me new shoes which impressed me.
I have a race coming up in 5 weeks and want to be in top form. I have wondered about weight training to build the calf muscle, but think that the extra work might strain them even more.
Anyone got any ideas on how I can strengthen my calves to avoid this problem? Or any other ideas welcome!