Why do I keep falling on my road bike?
Gravity.
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Am I just having a summer of bad luck? Maybe I'm not cut out for this.
It sounds like what you are doing is exceeding your level of experience.
First I wiped out on a group ride, I'm like 90% sure that a spoke broke on the front wheel the first week that I had the bike.
It's hard to say how this could have caused you to spill.
Second, I wiped out at the end of a long ride, less than a mile from my apartment on a path I ride on every day, but I don't know what happened because I don't remember it (concussion). Both of these were on straights, not in curves.
And finally today, again on a bike path that I ride on every day, I quite simply lowsided in a turn. The rear wheel lost traction, slid out from under me, and I have a nice big hole in my brand new Castelli kit. It's good pavement there, dry, I wasn't pedaling,
These other things could be that you are moving your center of gravity (CG) too far to the left or right for the speed you are travelling. And it could be partially due to the road conditions (maybe, there was sand that you didn't see).
Your road bike might require more precise control of your CG than your hybrid did. That is, the hybrid might be more tolerant of "sloppy" technique than the road bike. You might need to be less abrupt in your movements.
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What size tires are you using? You might fair better with wider tires (28mm if they fit). As you ride more, you can use narrower tires (but you might not want to).