Continuity mistake: When Allie's mother brings her back to the house after their drive, Allie's engagement ring is not on her finger. In the next shot when she is talking to Noah, her ring is on.
Factual error: When Allie goes back to see Noah and he asks her if she's hungry, he "catches" dinner. He picks up a crab net with about a half dozen blue crabs in it. Anyone who has ever crabbed, knows you have to pick the net up fast and keep pressure on the crabs or they will exit the net on the way up. Noah picks up the net so slow, there is no way he'd have caught any crabs in it. It is obviously filled with staged crabs.
Answer: She doesn't remember being married. Using her married name would only serve to confuse and upset her.