Welcome to road bikes! Both my commuter and race bike are road bikes and I love it that way. Yes, it is a different feel and yes, you should be giving yourself plenty of time to get used to it! After all, it's a big change. When I first got clipless pedals at the start of this month, I flipped over the bike about 3 times before I almost gave up. Thankfully, I didn't and I went to a quiet parking lot of a church and rode around.

I would suggest you do something similar because it will help build your confidence and help you adjust to it. Take some tight turns and practice looking around, stopping, starting, etc. It really will be great. If you don't already do group rides on the roads, I would also suggest finding a ride group. I have one and they are super friendly. Chances are the more experienced road bikers will help you as well. You'll learn all sorts of tips/tricks with them.

As for mirrors, definitely get one. I have one that attaches to my glasses. Many in my group have these and others have helmet ones or ones that fit on their handlebars. My preference is the glasses mirror, but everyone is different. It is a must for riding in traffic.

Making your bike as visible as possible, too, may also put your mind at ease. Sometimes it is more about you psyching yourself out than what is actually going on around you.