With freelance work you have quite a bit of control. The main thing is to hatch out what is and isn't your responsibility along with the pay. Write up a contract clearly defining your duties (number of drafts/revisions), deadlines, and payments (maybe right when you start and when you finish), invoicing will be important.
With easy small jobs/small pay, expectations will be managed, and the pressure won't be that great. It will acclimate you dealing with clients, which would still be a thing even when working in a company. You have to start building some experience somewhere, the confidence will come with it. What's the worst case scenario? If you try your best and they don't like it, it's not the end of your career.
Coding will probably be a big part of any web design job you will get, so you gotta weigh that against your current job satisfaction. I'm with Drak, consider getting a few freelance jobs under your belt. See how you like it and added to your resume can give you a step up in negotiating later. Wouldn't most web jobs give you more flexibility with work hours and commute if you can do a lot of the work from home?
If your wife feels overwhelmed with the kids, are there things you can do now to change things without finding a new job right away? Are these problems solvable more by money or time? is it like a new appliance or an extra hand putting kids to bed every night?
Do you have family nearby who can help maybe a hour or 2 a day or every other day (or a sitter) so she can run errands or just go out to unwind by herself?