The series starts in Atlanta and heads to the CDC located there before everyone walks through the Carolinas to Virginia. It takes multiple seasons for everyone to reach "Alexandria."
While they don't reach Alexandria until the last half of season 5, according to the fan wiki, they enter Alexandria on day 537 of the zombie apocalypse (less than 2 years), so most of the time they are in the south, it has not had that long to become overgrown.
And the Alexandria in the series is not the real life Alexandria outside of Washington, DC, which is a pricy suburb that was originally built in the colonial era, not the weird half-built housing development in the woods on the show.
In my head canon, it's another Alexandria, too, because I know it's a fairly dense area here. But in the show and the comic it is supposed to be the
Alexandria just outside Washington, DC. The houses we see in the show could easily fit in some neighborhoods of the real Alexandria. Here is a Google street view of a pretty upscale neighborhood in Alexandria:
Everything in the original series is in the middle of whitetail deer territory. Hunting season is a very big thing in both the filming location and the fictional locations because they are wildly overpopulated, contributing to agricultural damage, lyme disease via ticks and lots of car accidents, since the East Coast colonists long ago killed off the deers' natural predators.
Even if we assume that the zombies (sorry, "walkers") are too slow to ever catch deer (although many whitetail are so unafraid of humans that you can walk up to them and pet them, as I've done many times), none of the surviving humans would ever need to eat any meat other than venison.
You are right that there are deer around here, despite it being such a dense urban environment (more than people would think). With most people dead, gone, or undead, the deer population would grow, but that would take time. But predator populations would rebound as well. There are foxes, coyotes, and bears in Fairfax County next door, and feral dog packs might also spring up in the fall of civilization, so that might temper the deer population, but you are right that there should probably be a lot around. But if the deer population grows, that might help keep down the height of grass, no?