Congratulations on your new bike! A picture would be nice
I wouldn't say "just" road riding, it is all exciting and fun - with different challengesI am sorry to hear about your spill, that sounds scary. I am glad that you escaped without serious harm!
I got on a bike for the first time 6 months ago. That learning curve is an interesting one for sure. Have you considered putting a mirror on your bars? Personally I like Mirrcycle and they have one for road bars, it goes on the hoods. They do make several types for road bars and you will find that a good number of us here do use bar-end or helmet/eyeglass mirrors. I use my bar-end mirror to check what is coming up behind me, and then look over my shoulder to see what might be beside me.
Can you find a less busy route to work? I know sometimes that isn't possible. Perhaps give yourself a little time to practice on less-traveled roads before hitting the busy streets?
This will come with time, I have been riding for 6 months and can only now signal properly and drink from my Camelbak (not yet from my bottle). Can you do these things on your mountain or hybrid bikes? I assume that your hybrid bike has mountain bars? If so then your body is in a different position than you are accustomed.
For me at least, the key has been to not over-think/analyze things too much - this is my tendency. It doesn't workThe very best thing I have found - and very patient TE women kept drilling into my head - is to ride, ride, and ride some more.
I just took a Traffic Skills 101 class that was offered through the American Bicycle League and found it quite helpful - especially on how to maneuver through heavier traffic and dealing with busy intersections. There was also emergency stopping instruction and a few other things to help deal with unexpected situations. It was very inexpensive, perhaps it is being offered in your area?