Was Shakambari - one of the forms of Shakti - a rewritten story of Anasuya?

The Puranic story of Devi Shakambhari is as follows:

In old times, there was a great demon named Durgamasura meditated upon Brahma,  practiced hard penance for one thousand years.  Lord Brahma, became pleased with him, granted the boon Durgamasura asked for - Four Vedas.

From that time, the sages forgot all about the Vedas. So bathing, twlights, daily rituals, faith, sacrifices, and Japam and other rites and performances, all became extinct. Thus when great calamities befell on the earth, the Gods became gradually weaker and weaker, not getting their share of the sacrificial offerings. At this time, that demon invaded the city of heaven. And the Gods, not being able to fight with the demon fled in various directions.

Goddess Parvati, seeing the misery of all the people, out of pity, showered incessantly tears from Her eyes; and all the people and medicines were satisfied and out of those tears, the rivers began to flow.  

The Goddess gave them the vegetables, delicious fruits and roots to them that were on Her Hand, for their eating, until new crops came out. From that day She became famous by the name of Shakambhari (because she nourished all by vegetables, etc.).

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However, Srimad Ramayana, which preceded Mahabharata and all Puranas depicted a different story.

Sri Rama spends one night in the Ashram of Sage Atri, before entering into DANDAKA Forest.  There the Sage  Atri introduces his wife, Anasuya -  a great TAPASVANI by herself.  

(Ayodhya Kanda, Srimad Ramayana)

दश वर्षाण्य् अनावृष्ट्या दग्धे लोके निरन्तरम् || २-११७-९
यया मूल फले सृष्टे जाह्नवी च प्रवर्तिता |
उग्रेण तपसा युक्ता नियमैः च अप्य् अलम्कृता || २-११७-१०
दश वर्ष सहस्राणि यया तप्तम् महत् तपः |
अनसूया व्रतैः तात प्रत्यूहाः च निबर्हिताः || २-११७-११
देव कार्य निमित्तम् च यया सम्त्वरमाणया |
दश रात्रम् कृत्वा रात्रिः सा इयम् माता इव ते अनघ ||   २-११७-१२

"O, Irreproachable Rama! When the earth was burnt up by drought without break for ten years, this virtuous woman produced fruit and roots, caused the River Jahnavi to flow here, undergoing a rigid mortification enriched by pious observances, by whom the most severe asceticism was practised for ten thousand years and obstacles were done away with and by whom for the reason of a divine command, in a great hurry ten nights were reduced to one night.  This Anasuya, who has bathed after completion of the voes, is like a mother to you."

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So the writers of the Puranas (not Sage Veda Vyasa as commonly believed) perhaps copied this story and rewrote extolling the Devi Parvati in Devi Saptashati.