You could even out much or all of your backyard yourself. You could use a spade or purchase a sod lifter. The sod lifter would be much easier on your back. Take down the high spots and use that earth to fill the low spots. You can then seed the bare spots with grass seed if you choose. You could get a landscaping company to do it but I think you will be pleasantly surprised at how quickly this method will solve your problem.

I spy two coons! yes it's something you can do yourself. level with soil and plant the grass. With the tree gone now, you have a good chance.

I'm a bit confused by some of the comments. Is the second photo how it is currently or before you took out trees? Do you still have the black walnut? You seem to know about the black walnut not accepting other friends, but I'm putting a link in case it is an issue. Do watch that Virginia Creeper. It is native and can root like crazy anywhere it touches ground and drive you crazy. https://www.mortonarb.org/trees-plants/tree-and-plant-advice/horticulture-care/plants-tolerant-black-walnut-toxicity

Thank you! Yes, sorry for the confusion. The walnut will be staying - it's a beautiful tree. Thank you for the link. I am also dealing with a Bittersweet vine which is impossible to get rid of it choking out anything that grows in the back. :( Any tips on that would be appreciated.

You can level the ground as suggested but the shade from the walnut will make a good lawn very difficult. Grass needs sun.

One help with woody vines is to cut them at the roots and find something heavy to put on top of the roots. Like a big stone or brick. They are easier to remove when brittle and dead, which is what they will be after a few months of having no roots.