I honestly think the reason it looked like Carol wasn't covering T-Dog was the editing.
Now, I'm not normally one to throw the editing card around to excuse character mistakes, but in the scene where Carol and T-Dog rush the fence, the walker was FEET away from T-Dog before it cut to Rick, Glenn and Daryl. It made it look like Carol had plenty of time to shoot the Walker behind T-Dog, when in reality she probably had a fraction of a second. Walkers may be slow, but they aren't snails.
I'm not a huge Carol fan, but some of her hate -- just like the Andrea hate, is baseless and occasionally downright stupid.
Not to mention when it bites him you see an angle of where Carol would be stood and she isn't even there, and then she is when she shouts "No!", we are to assume she is just as busy and stuffs as hectic for her too, shes got Walkers near her too. If you were to ask the producers if it is intended to seem like Carol "failed" in her duty they'd most likely say no it wasn't the intention.
I though Daryl was very fast to assume Carol dead - but then again, he spent over a week looking for a ghost of Sophia, so maybe apart of him now believes it is best to assume the worst.
On a side note, another reason they showed the C-sec scene was to further push the idea zombies are now being used as tools and people are living in a world where they are trying to utilise them and make them apart of life. Season 3 is doing a lot of that, with Michonne/Milton as well.