Hi Nathalie,
I just had to respond to your post! I am a diver. I do enjoy it but like you, I suffer from the same fears! A few years ago I couldn't even stand by the side of the lake because its dark!
Anyway, a little over a year ago someone suggest I should try diving so that I could see whats under there and maybe I wouldn't be so afraid. So, I figured I would try it once and if its a disaster well, at least I tried. The dive was a success and I went on to do my Open Water.
Diving opened up a whole new world. I can now handle being on a boat (something I could not do before) and I even enjoy my dives. Yes, sometimes I have a little panic and there are times when I can't face going into the water. But once I'm in its amazing!
I have even gone to shark infested waters to see some sharks!

The first shark dive we did I was very busy NOT noticing sharks. But after a few dives I decided to be brave and take a look. To be honest the sharks kept well away from us. All that metal and bubbles and the noise... I don't think they wanted us around at all! The closest they came was approximatly 30m and they usually ignored us!
I would definatly recommend trying it out. Find an instructor you feel comfortable with and just go for it. Mark taught me and he is very patient.