For TSA they do not accept a temporary DL (REAL ID or not) so you can't just show proof that you've renewed into REAL ID. You would need to show other acceptable proof (or go through the no-ID process).
In my state while they don't always do it they often clip the existing DL and you'll need to use your temporary paper version as the other one is considered invalid once clipped. I have actually been denied purchasing of alcohol at a liquor store years ago with just the temporary annoyingly.
Just to clear that part up, please don't show up at the airport with a renewal application or paper temporary REAL ID without expectation of that potentially being an issue.
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