What a couple other guys have already mentioned...install the drip edge over the existing membrane, then tape the seam with an additional strip of membrane or flashing tape wide enough to cover the nails going through the drip edge and getting a good lap with the membrane on the roof. You can hit that with a roller for good adhesion.
Also, it's not completely It's not uncmmon on some installations to run the membrane past the end of the roof sheathing and down the fascia. Install the drip edge so the gutter (if there will be one) can be tucked behind the drip. The membrane on the fascia will isolate the fascia from the gutter.
If the edge of the roof sheathing overhangs the fascia by a bit, I'll go with a wider drip. If I'm not bending my own I'll buy something wide like an F5 or in some cases an F8.
FWIW, I first ran a strip of membrane on my fascia, up and on to the bottom part of the roof sheathing. THEN I installed the eave drip edge on top of that strip of membrane, holding it off the fascia a bit so the gutter could slide between the two. THEN I installed a full width piece of membrane on the roof, covering the drip edge. When I shingled, the bottom edge of the starter shingles overhung the bottom edge of the roof sheathing by about 1/4". The bottom edge of the first course of shingles overhung the starter course by about 1/8". Roof is due for a replacment, I'm now at 30 years. No negative issues with any of the detailing in that area.
Lemme find a picture...did find drawings, here are a couple of techniques from JLC. Adjust as necessary for your area and for the materials you are using. Just another version of how things could overap:
Eave Detail with Gutter:

Sidewall & Eave no Gutter:
Rake & Eave Detail: