What causes rainwater to follow the profile of drip edge and and run down the facia, rather than fall off the edge of the drip edge into the rain gutter?
Do your shingles extend at least 1/4" beyond the drip edge ? Did you overlap the first course of shingles (and install the bottom first row upside down) so that there are no gaps allowing water to leak down the fascia ?
I've seen cases where the water runs off the roof and when it hits the gutter nailing points, it finds its way back between the gutter and the fascia board at the penetration at the nail, and runs down the fascia.
Not sure if caulking around the nail penetrations inside the gutter (on the fascia end) will fix and for how long, but may be worth looking into.
Pushing the drip edge tight to the fascia board will cause this. Ideally as described above shingles will overhang drip edge by 1/4" or so. It seams many roofers find it easier to align with drip edge. Then with drip edge pushed so tight to the fascia board rain will just adhere to the drip and run right behind the gutters. I had to pull my recently installed drip edge out to bend it so the drip would actually act as intended. Now no more rain running behind my gutters. You should be able to gently bend the lower edge of your drip out and eliminate your rain running behind the gutters.
I used a 1" shingle overhang. You can say what you like about that dimension but its worked on my existing shop for 18 years and I can't detect an issue with the shingles. The drip edge in my case is more of a trim piece than a water deflector.
It happens all the way around, whether its a light rain or a downpour. I have not put any gutters on yet. I used a pliers with a broad tip to lift the drip edge at the facia and give it a good 'kick'. That seems to have helped, now I will do that all the way around. The roofing is IKO shingles, bottom course turned around. I set the shingles at about 1/4" back from the edge of the drip edge.