Installing drip edge under the two layers of shingle that should be present will take care of 99% of the water flow. The small amount that may come down during ice dam conditions may run under the edge. You'll need to decide whether that's a concern in your case.

You could also install your edge, then seal it with a narrow strip of ice & water shield. It'll bond with the existing layer over time and seal against nearly all leaks.

The bottom line is that anything is likely to improve your current situation.

It is possible that your ice & water shield will peel up cleanly depending on age, climate, current temperature, sun exposure, etc. Try that first, otherwise...

Installing drip edge under the two layers of shingle that should be present will take care of 99% of the water flow. The small amount that may come down during ice dam conditions may run under the edge. You'll need to decide whether that's a concern in your case.

You could also install your edge, then seal it with a narrow strip of ice & water shield. It'll bond with the existing layer over time and seal against nearly all leaks.

The bottom line is that anything is likely to improve your current situation.