What age are your kids? I’ve done the following non-Disney trips with my kids and enjoyed all:
Boston and then taking the Maine Coast up to Acadia is a great trip with a lot of history.
The Smoky Mountains area is a little kitschy, but I think the national park balances it out. You can easily get away from the crowds. Cabins are quite cheap in the area, too, due to high supply.
We stayed at the Great Wolf Lodge in Scottsdale, followed by Sedona, the Grand Canyon, and Tucson, which has a world class natural history museum / zoo hybrid.
Park City in summer. You have great resorts that are much cheaper in summer plus some traditional urban attractions nearby in Salt Lake City. Great outdoor recreation all around. PC is upscale without being snooty IMO.
Washington DC + Williamsburg is a good trip too, but I’d probably wait until kids are 8+.
San Diego is probably the most kid friendly city I’ve been to. I’ve always done it in conjunction with Disneyland, but you could easily do a week in San Diego and no Disney. Laguna Beach and Newport Beach are great, kid friendly Orange County beach towns - though Laguna is a bit colorful.