(snip)
when it rains, there is a tendency of water to drip down from between the
gutter
and fascia board in one section. I don't think the gutter is clogged and
there is no ice dam. I suspect there is just something about the way this
gutter was installed (or maybe the roof) that causes water to drip between
the fascia board and gutter rather than the water going just into the
gutter. There supposedly is an ice shield on the new roof.
(snip)

I agree this should be fixed, as it can lead to rot along the fascia, and
further down the wall, behind the stucco, brick, or siding.
Could be (1) a twisted gutter, due to loading or ladders or other damage, so
that the inner gutter edge is lower than it should be. Could also be due to
(2) clogging in the downspout pipe or the downstream gutter, or a
combination of of 1. and 2.
Either way, should be repaired.
Might be as simple as cleaning the gutters, downspouts, and releveling the
gutter, or reattaching the back of it to the roof or fascia.
Roger