Many thanks Monsoon - that did it!

Used "Navi" for camera movement. In "Characteristics of Preview", did "Lock", used Cubic projection instead of projecting texture from camera ("Cube mapping"). Now everything stays beautifully aligned as I rotate object and paint from my texture image onto my object - from various angles.

Yes, the fill tool has some nice features - most useful may be to fill a whole sub-object at once, so you don't have to paint it manually.