Before everything else, she was a survivor and would do what it took to survive, even if it meant begging.
Like I said to stormwreath above, it's a bit tricky to know what s1 Darla is capable of. We do know that in s2 of Angel, it takes a long time for her to come to grips with the idea that she's dying, and she's prepared to do just about anything to escape it... I certainly don't think it's a stretch that she'd try begging for her life, just like she ended up begging for her death from Angel.
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Before everything else, she was a survivor and would do what it took to survive, even if it meant begging.
Like I said to