That IS the big problem with trying to do a zombie show that's long lasting. That's why the majority pretty much show the initial outbreak, or a short time thereafter, so you can skip a lot of the logistics that drag you down later. Zombies only really make sense in a short-term way.
So to make the show work, kinda HAVE to be stuck in a perpetual '1 month after outbreak' timeframe.
Otherwise, the zombies die/fall apart, the human defenses get too built up, zombies become commonplace enough that no one's afraid anymore, etc.
You NEED the fresh threat of zombie herds, and the certain amount of fright or at least overwhelming numbers, or the concept just doesn't work. Then it's just random post-appocolypse show, and whenever anyone dies, you stab them in the head and move on with your day...