This is not the answer you want to hear. A few years ago I had a similar problem. I bought a book at Lowe's, bought the materials, cut out the rot, carefully did all the repairs, priming, painting......
I would have saved money if I had just switched the hardware to new shutters. Even if my time were worth $3/hour I would have been better off buying the shutters instead of the book. Don't throw away the shutter hardware -- it's expensive. Good luck.