Catchingup,
That's a normal feeling! I was terrified of becoming a mother, and then I became one and was even more terrified. Then came the time for daycare, and it only got worse. I think the idea that something could go wrong is something we need to learn to live with. Everyday I get up, I am so glad I made it through one more day to spend with my DD, without something happening to either one of us. I don't think the fear that something could go wrong will ever completely go away, but that doesn't mean that we need to avoid the joy that can be in the present.
Terrible things happen all the time, and while I certainly wouldn't want to be the one responsible for anything like that, there are some things that are just beyond our control. I read a story about a dad leaving his son in the car at work all day in the hot sun. I was terrified I or my DH would do this. I called the daycare everyday to make sure DD got dropped off. However, I FINALLY looked up the statistics; this happens in this big country of ours (U.S., assuming!), about 5 times a year. The odds are so miniscule, that we simply have to live with the fact that it is an imperfect world, and sometimes even if things appear within our control, they actually are not.
The fact that something could happen is exactly why you NEED to be an integral part of your GCs lives (especially if your kids allow you that opportunity!). I lost my best friend of 20 years to cancer last year, and we had lost contact for a few years before then. I decided to reach out, and shortly after that she was diagnosed. I am so grateful for that precious time with her, even if it was spent in a hospital, and watching her die. I think her and I got so much out of just loving despite the fact that things might go wrong. Your GCs need to know who you are because we just never know when our time is up. Sometimes the bad times can be just as enriching as the good, and sometimes good things do come out of tragedy.
I'm learning to live happily despite having the fear of something happen; and I'm only now just learning what true love can overcome. Don't short change yourself or your GKs out of fear.