I walked the CF twice at this time. The first time, we had a week of sun at the beginning, then it rained most days for the rest of the walk. The second time, it rained the first three days, and we had glorious sunshine for the balance. I carry TWO pairs of shoes, Gortex trail shoes and running shoes. I don't recommend this for most pilgrims due to the excess weight, but for me its ideal since my feet appreciate a shoe change from time to time.
On balance, if the weather is good, you are best off wearing your favorite running shoes, or lighter trail shoes. If its raining a lot, not so much, because heavy mud can become an issue in spots. The trail itself does not demand a heavy shoe or hiking boot, but your feet may. Check the long range weather a few days before you go, and make the call then. Wear the one you find the most comfortable for the expected conditions. Bear in mind you could take your heavier gortex shoe for the first few weeks, and pick up some runners in Leon if the weather improves.
Buen Camino