in order to correct the slope would it be enough just to reposition the same screws into new holes just above or below the existing ones?
Somewhat. If the gutters were properly installed, the facia brackets are likely mounted with 1/4" lag bolts. Depending on the length of each gutter section, you may only be moving these bolts up or down a very short distance. I'd recommend picking up some Liquid Nails and/or a tight-fitting dowel to permanently fill the hole left behind by the old bolts. That way there's little chance of sagging down the road from collapsed bolt holes.
the corners are made like in the picture below. I am affraid that while moving the joint gutters I might crack the sealing, what do I need to do to fix it properly ?
This will really be a case-by-case thing. At a minimum, you should pick up a high quality gutter sealant (Such as GS121) and use it inside the gutters along the seam. Some gutters are installed using only the strip miter and sealant, others have pop rivets, and some have these fancy, yet mostly useless, corner slips. Lacking manufacturer recommendations, there's not a right or wrong way to do this. Though, in my experience, high quality sealant plays the biggest role by far in the watertightness of your corners.
