How to fix bunged up shingles &drip edge

While you’re on roofing questions… I’m a new homeowner, and though I do a lot of woodworking, I know very little about carpentry. I recently bought a home and noticed that the rain was running down between the facia and gutters on about 1/3rd of the house. I got out the ladder and found that the roofer didn’t extend the first row of shingles down to the end of the roof sheathing in some spots. Also, it looks like they used bent flashing instead of drip edge. It’s top edge is rounded over as a result and this adds to the problem.
How do I fix this without re-roofing the house? Will it work to simply pull off the gutters, facia, drip edge and then install new drip edge and then shove some cut shingles under the first row and glue everything down with shingle or NP3 caulking?
I’m in a coastal location (Houston, Texas – home of the slap together on a slab neighborhood) prone to hurricanes. The roof is only 5 years old. We bought the house new and man did I get sick of the phrase “…this is not a defect.” in the structural warranty. At any rate, I’m probably going to redo all the soffits and facia but have no knowledge of how to do the shingle/drip edge juncture to where it isn’t going to leak down the road.
Replies
Can you take pictures to show us?
It sounds like a mess, exposed roof decking?
The shingles in the problem areas fall short of the roof's edge by not more than 1/2".
I would use a drip edge like this profile. How much too short did they leave the shingles?
http://www.eastcoast-metals.com/fha.html
Wow, real drip edge. Wish I had some of that on the house. The shingles fall short by at most 1/2". I guess the OSB sheathing would be exposed except for the tar paper and bent over flashing.
I can't believe they did this. But then again, I had to call the builder back after I noticed they'd simply shingled over the cut in the roof for the ridge vent and forgot the vent itself. I also didn't know they made one-month caulking either. There's not a single caulked joint that didn't peal away within 6 months.
If this is a new house, I would have a lawyer contact the builder about this pure negligence and forget doing it yourself.
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Too late for that now.
The facia boards are pine that were only sprayed from the outside. The inside (I checked from the attic) and jointed edges are bare and they're starting to warp and rot. I think I'll cut and paint enough cedar boards for new facia and then call in a local (reputable) roofer to come in and fix this.