Karwa Chauth is mentioned as करक चतुर्थी in Hindu scriptures like Vāmana Purāna, Nārada Purāna, Nirñaya Sindhu, Dharma Sindhu करक is करवा (water vessel)
कार्तिककृष्णचतुर्थी करकचतुर्थी। स चन्द्रोदय व्यापिनी ग्राह्या। दिनद्वये तदव्याप्त्यादौ संकष्टचतुर्थीवनिर्णय:।
Here is an excerpt from part of Nārada Purana where Vratas performed on Chaturthi is mentioned. Karaka Chaturthi (Karwa Chauth) is described as: There is no other vrata like this that yields conjugal blessedness to women than this.
It describes a Vrat:
Karak Chaturthi, celebrated on Chaturthi (fourth day) of Krishna Paksha (dark phase of Moon) of Kartika Month which is the Karwa Chauth day. Arghya gave to Moon at the fulfillment of Vrat. Presents given to Suvasinis (Married women).
The third reference is from Vamana Purana which describes the day for the celebration of Karak Chaturthi i.e. Chaturthi of Krishna Paksha of Kartika Month. Women worship Gauri, Ganesha, Shiva, and Shanmukh (Kartikeya) on this day. Even on Atla Tadde, Gauri Puja is performed.
There is also mention of what is thought to be popular folklore of Draupadi performing Karwa Chauth. When Arjuna goes to Indrakaal parvat for penance, Draupadi is tensed. Then Krishna guides her by narrating story of Lord Shiva narrating Karwa Chauth's Katha to Maa Parvati
Then Lord Shiva narrates the story of Veervati to Mata Parvati. Veervati who was tricked by his seven brothers to break the fast before Chandra Udaya due to her health issues. Later on, Goddess reminds Veervati of the trickery & she performs a strict fast next time & regains her husband.
Atla Tadde celebrated in Telugu states i.e. Andhra and Telangana is the just the same festival celebrated a day before Karwa Chauth. As Atla Tadde is celebrated on Tritiya (3rd day), probably because of the difference of calendar and festival is linked with Gauri Puja/Gauri Tritiya.
Just like there are Atla Tadde (3rd day) and Karwa Chauth (4th day).In Jammu Kashmir, Kashmiri Pandits celebrate Gauratrai (Gauri Tritiya) in Magh Month and Bhaderwahi Hindus in Jammu Hills celebrate the same festival the next day as Kan Chauth.
In some parts of Rajasthan, there is a tradition to keep fast on almost every Chauth (Chaturthi). Goddess Parvati is worshipped as Chauth Mata. Chauth Mata, Karak Mata, Karwa Mata, Mata Gauraa are all epithets for Goddess Gauri.