Wow, another Madisonian! Have you considered trying to rope your wife into the trip? That way you have two adults pulling two tag-a-longs, and everybody wins! Additionally, I BELIEVE there is a campground out in Lake Mills which is accessible via the Glacial Drumlin state trail. However, you would need to take Cottage Grove Rd to get from Madison to the trailhead in Cottage Grove if you weren't up to driving out there.
Additionally, you could look into the Badger Trail which goes southbound out of Madison. I don't know of any campgrounds in that direction, but only because I haven't looked. Apparently there is a 7 mile uncompleted stretch of the trail at some point before Belleville, however.
Lastly, you could look into taking the kids up to Devils Lake State Park. I don't know that there are any trails that'll take you there, but it looks like mostly quiet county roads, so not too tough to have the kids on.