Not familiar with the term "drip cap." Are you referring to aluminum cladding which is "wrapped" onto the fascia?
If you mean drip edge, gutters should not be attached to that at all, nor are they normally forced up behind it. They're set below the lower edge, and under normal circumstances, water will literally drip from that edge (if not from the more pronounced one above) into the gutter.
As for adding nails to better secure the drip edge to the roof sheathing, you should be able to lift the lower edges of the shingles enough to do so without damaging them.
If you mean drip edge, gutters should not be attached to that at all, nor are they normally forced up behind it. They're set below the lower edge, and under normal circumstances, water will literally drip from that edge (if not from the more pronounced one above) into the gutter.
As for adding nails to better secure the drip edge to the roof sheathing, you should be able to lift the lower edges of the shingles enough to do so without damaging them.