Having a cart full of items, Anya let the woman buying just one thing go ahead of her in line. Having a cart full of items, the woman buying just one thing moved ahead in line.

The first sentence makes sense; the second one doesn't.

Explanation:

Having a cart full of items

is a subordinate clause that makes sense because it refers to the subject Anya in the principal sentence

Anya let the woman buying just one thing go ahead of her in line.

Because Anya has a cart full of items, she decides to let the woman, who is buying just one thing, go ahead of her in line.

In the second sentence the same subordinate clause doesn't coincide with the subject

the woman