Fragility

Why the hell am I so fragile? Why isn't 24 hours enough to recuperate from 30 minutes of running? I've been running 2-4 times each week now for little over two months, I have proper shoes fitted by a specialist, I run on a soft surface (track), and I still end up with sore muscles after each run. I need 1-2 days of recovery in between sessions to not break down completely, and it's pissing me off!
You would have thought that my body would have adjusted by now, but noooooooooo. I'm doing everything by the book, yet I still break my body each time I lace up. I's incredibly frustrating, especially since there doesn't seem to be anything I can do to get better. While my stamina has increased (5km in 30 minutes, with some walking inter-spaced), I can't seem to get fit enough for mu muscles to handle this simple thing of a run a day. I'm not asking much. I'm not asking for hours each day, or a triathlon each weekend, just grant me 30 leisurely minutes of running each day, is that too much to ask?
Here I am with an ice pack on my shins, having to cut my run short after 20 hellish minutes before admitting defeat in the face of having to spend 1-2 weeks not running at all should I continue and completely trash my legs. Today it even got to the point where my thighs started hurting. My thighs! My thighs never hurt, normally. It's always my calves (less so lately, however), and my shins (always a fucking pain). I stretch and I do toe-raises as my desk daily, but that doesn't seem to help the fact that my shins never get with the program. I guess I could spend 2-3 weeks completely off running and physical exercise to let them heal, but then I'm back to square one again. I will have lost my stamina, my remaining muscles will have weakened, and I have to start building them up again, which inevitably will kill my shins.
It's a vicious cycle, and I don't know how to break it.
Help me Obi-wan Kenobi, you're my only hope!