So it was originally designed to wash the window every time it rained.
That post may have sunk but I don't think that's the main cause of the sag. There is just too much eave overhang, without trusses or rafter ties (no ceiling joists), the rafters are just not stiff enough. Especially that barge rafter on the end, it only has a partner on one side. This is a common problem in my neighborhood where many of the houses have truss roofs but a similar long overhang. The houses with a gable at the front have 3 corbels positioned in about the same places as the beams in this house. Side and rear gables do not even have the corbels.
Even with corbels the rafters, especially the barge rafters without truss support, sag. Some of them have bowed over the years causing a hump in the roof slope. Some have sagged in the middle making a dip in slope and a kick at the end, like a ski jump. One of my barge rafters even broke late one night.( Some folks awaken at the crack of dawn, the crack of a rafter will really get you out of bed.)
Jacking up the post high enough to cure that sag could put a hump in the roof, especially if the rafter has bowed. If 'twere me, I'd put a post right out at the corner. A 4X4 or 4" pipe on proper footing would probably do. Or, if ya just don't want a post there, possibly a stiff horizontal beam back to existing post and on back to wall. A hollow corner post could replace that weird down spout. Or use the existing spout with the C and hang a loud wind chime on it to annoy the neighbors.