Fascia, drip edge, starter shingle, code, in Texas
Laying down a roof starts from the bottom and goes toward the peak. Trying to repair something like the drip edge or lowest run of shingles is almost never going to give a good result without starting over. 

If you need to replace the fascia board, you mentioned rotting, usually also involves stripping the roof, especially when there is an existing metal edging installed over it and maybe even nailed into it.

In my experience in working on roofs hanging the shingles a small amount over the drip edge does not significantly increase the ability of wind to rip the shingles off the roof. Of course it is best if the roof is laid down with the very first course using a starter roll instead of shingles. Good quality starter roll has a row of tar strips a short distance up from its lower edge to which the tabs of the first row of shingles are meant to bond.