Becca,
I had taken on some similar goals and am also new at road biking (just bought my first road bike on April 13). I had signed up for a 10K run on June 23, a 500-mile bike ride July 15-22, a sprint tri on August 18, and a half-marathon on September 16. My longest run to this point is 6 miles. I can swim more than a mile comfortably.
Finally, I realized that this collection was insane and that I needed to prioritize. I've given up the half marathon, and I changed the 10k to a 5k (and if I decide not to do it that day, no problem). So that leaves me the 500-mile ride as my training priority, with the sprint tri a month later.
I am concentrating on biking, and started at 10 miles, increasing 5 miles/week. Monday, I'll do 45 miles. I try to do three rides a week, at least, with one of them being the long ride, plus I've been doing a group ride of 20-25 miles for the past 3 weeks.
I try to run at least once a week - just 5k. When I get back from the big ride, I'll increase that to twice a week, and will commit to resuming swimming at least once a week. (I'd like to swim once a week now, but I'm not going to obsess over it).
I still feel a little bad about giving up the half marathon, and if I really want to do it when I'm done the 500-mile ride, I'll do the training... but I don't love running like I love biking.
Could you lengthen one of those bike rides each week? If you did that, you'd be just FINE. Just increase it by 5 miles each week. Or even less.
Good luck!!
Suz
2012 Trek Madone 4.5 WSD
2007 Trek 7100