To be honest I don't think you'd ever stop feeling like that no matter which roads you take. I worry every time I venture out with one or both of the kids on the various bike attachments - I think that is a healthy mindset and helps to ensure you do everything you can to ensure a safe journey. There is only so much you can do though, given how some people use the road, so you have to decide if taking your kids on the road feels like too much of a risk to bother with at all.

Happy to let you try out the seat and/or trailer if you want to see how they feel (although the seat needs a Blackburn EX1 rack so you'd probably have to try out one of my bikes for that).