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I see 2 problems. Both can be solved.
Position of the camera sensor plane parallel to your object's plane.
Place a sheet of squared paper on the table for the test shot. So you can see the proper alignment in both dimensions at once. Use a tripod to fix your camera above the table.
Tell the camera to insert a little time delay (3-5 seconds) when shooting to eliminate the motion from your finger's push.
(Edit: if you really want to stay very cheap, you could try without the tripod. But use the squared paper behind your object. Then do it all in the post processing).
Lens distortion.
Use a lens of a well known brand.(I don't know if it is possible with a phone instead of a real camera). Then use the software "darktable" for post processing, because it has a library of lenses and can properly correct the distortion with 1 click.