I think the quickest way is with JOSM + the Building Tools plugin. If the buildings are all rectangles or "L" shaped then the Mapathoner plugin is really quick too.
Building tools is quick enough that if I'm fixing an area that's been mapped wonky it's often a lot quicker to re-trace the skewed buildings in a fresh layer and use the Conflate plugin to merge them back into the main layer than it is to fix the geometry directly (and that include individual checks).
In terms of keeping track of which areas you have done and which you haven't I think SimpleTaskManager is really good for single person projects. Divide and Map Now looks like it should be good for more collaborative stuff but I haven't tried it in a while.
https://wiki.openstreetmap.org/wiki/SimpleTaskManager
https://wiki.openstreetmap.org/wiki/Divide_and_map._Now.
Unofficial Canadian OpenStreetMap community.
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Apps:
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StreetComplete: Surveyor app where you can complete quests
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