How to fix my roof draining into gutter improperly
I agree with your assessment that these corners are getting wet, and it is because they are improperly formed.
As commenter suggested, the gutter typically goes all the way to the roof edge, as in this picture (crudely edited from what you posted). This would require cutting the extra-long fascia shown in red.

Alternatively - much easier, but kind of a stop-gap measure you can use &quot;in the meantime&quot; while you prepare to cut fascias/extend gutters - you can at least stop the water from getting in here using another piece of aluminum flashing that extends from under the shingles to over the end of the fascia. The key here is to route the water from the roof around the top of the fascia where it has been sitting long enough to seep under the paint. Note that even with this measure, you'll need to properly address this soon because the fascia wood has already started to rot.