If your shingles are in good shape and flexible, you may be able to pry them up enough to pop off the old drip edge, and slide the new into place. The tricky part is to nail, or staple, it from above without damaging the shingles.
More likely, you will find it easier to slide the new drip edge in place over the old and staple it in place on the facia.
Either way, this is much easier done without the gutter in place.

Had a similar issue and decided to extend the drip edge in the following manner: loosen or remove entire length of gutter (I loosened mine rather than removed so as to aid in the gutter re-install process because this was a one-man job). Cut galvanized or SS flashing (purchased by the foot @ Ace) into manageable lengths 6"x48". Pried drip edge away from fascia w/ wide putty knife and installed flashing between drip edge and fascia. Sandwiched flashing behind existing drip edge and in-front of fascia. Fastened drip edge and flashing to the fascia w/ screws and over-lapping flashing srips the entire length of the fascia board. Re-install gutter w/ screws and w/o any sag while allowing gradual slope to drain. This worked well for me. Hope that helps.

On long gutter runs, the gutter will drop below most drip edge due to the slope. Sometimes you have to use trim sheet and fabricate a flashing to fit behind the gutter lip.

Michael

Thanks...
What's a trip sheet?

I'll need to see if I can loosen the gutter sufficiently to give me some working room without making the whole assembly collapse!

Also, I think I'll need to check on some of the Fascia to see the extent of damage from years of exposure to water (due to a clogged and sagging gutter).