No - they will not short just because they cross each other if they are on different copper layers.
Simple rule:
Same layer + touching/overlapping = likely short
Different layers + no via connecting them = no short
The fiberglass/core/prepreg between layers insulates them
So in your example:
Top Copper
Mid-layer 1
Mid-layer 2
Bottom Copper
Those can pass over the same XY location and still be totally fine, as long as:
there is no via connecting them
there is no exposed pad/copper structure intentionally tying layers together
your board stackup is valid and manufactured correctly
What can still happen on different layers:
not a short, but possibly coupling/noise/interference, especially for fast digital, RF, clocks, or sensitive analog signals
So for your current error: if Flux says Overlapping Copper, it usually means the overlap is happening on the same layer, or through something conductive that actually connects them.
If you want, I can explain your exact Trace d0a8 vs U4:VN error in one line using your board.