When I have a shape's fill set to a pattern in my swatches, and I move the shape around, the pattern doesn't move with the shape. How could I make the pattern offset be relative to the shape (Move with the shape, so that moving the shape doesn't mess up the pattern), rather than the artboard?
This should should be your solution in Adobe Illustrator: Go to
Settings > General and check
Transform Pattern Tiles. Be aware, that this also scales the Pattern when scaling the shape.