Unfortunately, the best way to find the big swamps is to go out farther. There are plenty of black forest and even meadows out farther, you just have to watch for them.

Try sailing out father from the center and first looking for black forest (or meadows) that you can land on to establish yourself out there, then explore from there, making sure that you can get back there if a big storm comes in or before night. You'll probably see other safe areas to land while you're exploring that you can go to if you get in trouble.

If you haven't used some of your iron to make a longship, I highly recommend it for such travels. It's much safer and easier to sail in the open ocean.