Correcting a possible roofing error
On the roofs I've done, the shingles (or metal, in one case) extended 1/2" to 1" past the flashing.  It does not sound like this is the source of the problem for you.  The flashing should protect the fascia from a full water flow, if it is the correct flashing.  You describe it as L shaped, but it should have a kick out at the bottom ("drip edge").  I found a decent picture here.

In addition to a small overhang and proper flashing, the final defense for your fascia is a coat of paint.  If the fascia is not rotten, you should restore its surface, and prime and paint.