since all posters seem only interested in calling you out for adding connections to your flight I will address the question asked about SDC with a first class fare.

You can SDC to any new flight that is a valid routing as long as there is at least 1 seat available in the First cabin. Same fare class is not needed like it is in economy.

If you get push back on this from delta agent hang up and call again. I don't have time to find the link but there is clear language on delta .com explaining this SDC rule for first class pax.
Quick search I'm sure you can find it easily.

Good luck