There are weird shifts in the texture and I don't know how to fix this
I tried to export it without the UV maps from blender but that didn't help.
If you know how to fix this I would like to hear it :).
(The frog is for reference for what is should look like.)
Your UVs are not correct, you need to fix them. You have several UV islands that overlap each other (see the circle shape overlapping the rectangle in the 2D View), which lead to what you are seeing.