I considered the rain cap, but it doesn't address water intrusion behind the trim that it attaches to. You still have to rely on screws run through butyl tape to not leak. I ran 4" eternabond from the top of the roof into the gutter itself, covering the entire piece. No chance of water intrusion.
To the OP, you likely have a low spot in the gutter. Other than try to bend it up just a bit. Might be a good idea to get on the roof with a water hose and see if you can see what's actually going on.
Mine actually had a seam in the gutter that wasn't sealed very well. And of course, it was dripping right above the door and giving the exact same issue you are having.