I get a JSON Response from my API. The results save configuration window only shows XML, and the csv generated does not have any response data.

Does this not work with JSON responses?
Generated the CSV, got these columns only:
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If you want to switch JMeter's .jtl results file to XML mode and see response data in there you need to amend JMeter Results File Configuration and add at least the next 2 lines to user.properties file (lives in "bin" folder of your JMeter installation:
see other properties and enable saving of those you're interested in. However be aware that it violates JMeter Best Practices because it causes massive disk IO and can impact or even ruin your test.
More information: How to Save Response Data in JMeter