We have a sawyer on the website that once cut up some gum and stickered it for air drying. It became totally useless as it twisted and buckled as it dried. I'm sure there are others here that can give you first hand experience with attempting this.
It can be a decent turning/carving wood, but it is hard to get straight boards out of it. It moves around like crazy, even after it is dry enough to use; seasonal movement is pretty extreme (it's about as close to migratory as wood gets ). The interlocking grain makes it very difficult to split and it doesn't burn all that well, so it is so-so firewood. It can be really pretty, but it is one of the few species I don't regret seeing chipped for mulch. Our forefathers made a lot of treen with it; a wood that doesn't split easily is great for kitchen utensils and similar "one piece" uses where a little movement is hardy noticed.
It is a very challenging wood to dry flat. I used to sell lumber for a mill in Louisianan that cut and dried sweetgum. The white sappy boards were sold as sapgum and the red heartwood was sold as red gum. You can dry sweet gum flat if you put tons of weight on top of the pile (sticked lumber) to reduce the twisting of the wood as it dries. You must get the lumber into a kiln very soon after cutting as the sapwood will stain quickly. Once dried the boards that remained flat had a good markets. Most of the sapgum went to Italy where they stained it to look like cherry and most of the red gum went to California for renovation molding (to match stock originally used back in the late 1940's). The red gum brought prices very near walnut and made beautiful panels for some kitchen doors I used to take to trade shows. We couldn't make enough red gum boards.
I do recall an article on the news awhile back about a company in Plains, GA that is actually using sweetgum to make chopsticks and exporting them to CHINA! They are making over 1 million per day IIRC.
The more I research this wood the more I think I would be better off letting the Arborist haul it away when he leaves. It's difficult to split, is only rated fair for firewood, and it doesn't make good lumber.
I tried the same thing and it really twists and shrinks quite a bit. Once dry I got a lot of short straight boards that I often laminated together to use. It finishes OK and has interesting grain especially near the trunk. Good for painting or secondary wood.