I’m having trouble seeing the discrepancy (“out of plane in the second plane” ) with the photos that were provided. The photo with the blade parallel to sides of the mouth is clear, but not the other.
Keep in mind that a perfect plane will move the edge out of square depending upon how it is used. When learning to plane, it is common to unintentionally apply downward pressure, especially if the plane isn’t really sharp, or to apply a twisting or rocking force to the plane, which will lead to changing the squareness.
There are planing methods that will remove the tilt. When I reduce an edge, I typically remove the bulk and then adjust back to square because big adjustments do tend to brng the edge out of square quite easily.
So, there may not be anything wrong with the plane….