I'm not a grammarian, so let me just answer your question in plain English.
In your examples, "get my shopping done" and "do my shopping" mean the same thing. "Get done" is being used idiomatically to mean "finish" or "complete," so you're using it the same way you're using "do" in the second example.
I don't know whether that's the causative form. Let's see what others say.
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