I would recommend a gait analysis at a proper running shop.
I had mine done and as a result of the analysis, I was recommended Brooks Adrenaline GTS shoes as they provide good stability to correct my apparently unstable feet and ankles!
Had I not have had the analysis done, like most people, I would probably have just set a budget and bought whatever Nike/Adidas shoes came in at the right price. I would certainly never have brought Brooks trainers.
Anyway, all this waffle may or may not help you, you might need to stretch more? Daily rather than just before/after exercise?
You might just be overdoing it and need to lay off for a while?
One of my running buddies who suffers with sore calfs/shins, swears by compression socks (the ones that come up to the knee) to help him.
I'd say get your gait looked at and treat yourself to the 'right' shoes, take it a bit easier for a couple of weeks and work from there.
I'll add a bit more... if you're only jogging at the gym for exercise and not because you actually like it or want to run a 10k or whatever, then why not just stop altogether and substitute a different exercise that doesn't stress your lower legs like this? Plenty of other ways to improve your cardio performance and lose weight without pounding the f*ck out of your legs three times a week