I don't know that there's a right or wrong approach to any given hill. Some people do better spinning in a smaller gear. Others do better staying in a bigger gear but slowing their cadence down. Sometimes, your best bet is to do a little bit of both.
The nice thing about a long climb or a false flat, is that it's easy enough to play around with your technique. So, that's what I would encourage you to do for now. Next time you hit a false flat, stay in the same gear and let the ride slow your cadence down a bit. If it gets too slow and/or your knees or quads tell you "no thanks," shift to the next easiest cog. Or do just the opposite. Shift to the next easiest cog, but keep your cadence the same as it would be on the flats. Don't worry about getting passed by somebody. You don't really learn how to pace yourself up a climb if you're not riding YOUR pace.
For me, the most important thing is that I feel in control. My bike and breathing are steady. My upper body is relaxed and my chest is open. My cadence is controlled and even. My mind is staying in the present (I'm not impatienct about getting to the top).
Live with intention. Walk to the edge. Listen hard. Practice wellness. Play with abandon. Laugh. Choose with no regret. Continue to learn. Appreciate your friends. Do what you love. Live as if this is all there is.
--Mary Anne Radmacher