“What can we do to keep the Polk weed from coming back AND get grass to grow this spring.”
I think with Polk Weed, you need to nuke it from space! ;)
Unless I misunderstood, getting grass to grow around trees/in shade is near impossible. Once you get the Polk Week out, (everyone hates Round Up, but this is probably a job for Round-Up) I’d either go for a ‘Shade Garden’ or plant some sort of ground cover that likes the shade.
More information would help me help you. Did you also clear the trees completely? What’s the sunlight going to be like where you want to grow grass? :)