Answer: You could use a shovel, but it would be labor intensive. It would simplify things if you rented a rototiller and tilled the entire area up, raked out all the old grass and roots, then leveled it and broadcast seed. This would allow your new lawn to get a fresh start and you'd end up with a thick, lush lawn without established weeds.
There's another approach you might consider, though. Start by eliminating all the weeds - either digging them up roots and all, or applying a crabgrass killer and raking out all the dead crabgrass. Once you've raked out all the weeds, spread top soil over the low spots to make the whole area level. Then broadcast seeds over the entire lawn. Grass seeds will germinate in 7-10 days and if you keep it well watered you'll eventually have an almost new lawn.
Best wishes with your lawn!
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