Helene, it seems like there is some anxiety surrounding riding for you; I can actually relate, and I've been riding a long time. Getting a flat is part of riding. Learn how to change a flat, and carry the appropriate tools, so even if it's hard, you have the tools and someone could help. Try visualizing yourself riding, carefree, down the road. Do some deep breathing before and as you start riding, and find a mantra to say to yourself while you are riding. In fact, I think we had a thread about mantras for riding somewhere.
I did find that the 25c tires made a difference in my feeling of stability; it doesn't seem like 2 cm would make a difference, but it did. In the end, you may have to find a different bike to feel comfortable. And stay away from shop guys and racer types that might push you in a direction that you don't want to go.