As mentioned in the comments Shankhchuda was an incarnation of Sudama not Vishnu. The Shiv Purana section 1.2.130 in the shared link mentions the following story:
1.2.130 Birth of Shankhachuda - Sutji narrated the tale of the birth of Shankhachuda and how Shiva killed him with his Trishula. He told the sages that Shankhachuda was born to the demon king 'Dambha'. Shankhachuda was in fact, Sudama, in his previous life. He was born in the family of demons due to the curse of Radha.
1.2.131 Shankhachuda’s Marriage - When Shankhachuda grew up, he went to Pushkar and did a tremendous penance to please lord Brahma. Lord Brahma blessed him and said that he would remain invincible. He also instructed Shankhachuda to go to 'Badrikashram' where he would find his would be wife Tulsi- the daughter of Dharmadhwaja. Shankhachuda went to Badrikasharam and married Tulsi as per the instructions of lord Brahma. He then returned back to his capital accompanied by his wife- Tulsi.
The rest of the story goes as follows:
1.2.132 Shankhachuda Becomes the Ruler of All Three Worlds - When Shankhachuda reached his capital after marrying Tulsi, he was crowned as the king of the demons by Shukracharya. After his coronation, Shankhachuda attacked Indrapuri supported by his huge army and defeated the deities. In a very short time all the three worlds were under his control. After being defeated by Shankhachuda, the deities went to lord Brahma and sought his help in eliminating the menance called Shankhachuda. Lord Brahma then took them to lord Vishnu. All of them were taken to lord Shiva by Vishnuji. They expressed their request to liberate them from the troubles created by Shankhachuda.
1.2.133 Shiva Assures the Deities - Lord Shiva assured the deities that he will certainly kill Shankhachuda and hence they should not worry about him. The deities then happily returned to their respective abodes.
1.2.134 Conversation Between Pushpadant & Shankhachuda - Lord Shiva sent his messenger named Pushpadant, to Shankhachuda to ask him to return back the kingdom of the deities to them. Shankhachuda refused to oblige this demand, on the contrary he expressed his willingness and readiness to fight lord Shiva. Pushpadant returned back and narrated the whole story to lord Shiva.
1.2.135 Shiva Proceeds with his Army - Lord Shiva was now fully convinced about the inevitability of the battle. He first sent all his ganas under the leadership of Kartekiye and Ganesha. Later on Bhadrakali proceeded towards the battlefield with a huge army as per the own wish of lord Shiva himself. At last lord Shiva proceeded towards the battlefield accompanied by the deities. All the troops collected at the bank of river Chandrabhaga and rested under the shade of a Banyan tree.
1.2.136 Shankhachuda Proceeds with his Army - Shankhachuda handed over the kingdom to his son and went to his wife to take her permission before going to the battlefield. His wife was reluctant to allow him to go, but he somehow managed to convince her. He then proceeded towards the banks of Chandrabhaga with a huge army.
1.2.137 Battle Between the Deities and the Demons - A fierce battle started between the deities and the demons. Both of them attacked each other with the most destructive weapons. But when the army of Chandrachuda began to dominate the fight, the deities fled away and took the refuge of lord Shiva. They told Shiva of the number of deities who had been killed while fighting.
1.2.138 The Battle Between Shiva’s Family & Shankhachuda - After the defeat of the deities and the ganas Kartikeya and Ganesha went to fight Shankhachuda. A spectacular battle was fought between Shankhachuda and both of them. Later on they were joined by Bhadrakali. Bhadrakali would have devoured Shankhachuda without any problem, but she spared his life because of the boon, given to him by Lord Brahma. Now it was the turn of Lord Shiva to join the battle, but even he could not cause any harm to him because of lord Brahma's boon.
1.2.139 Strategy Employed to Kill Shankhachuda - While a tremendous battle was being fought between lord Shiva and Shankhachuda. Lord Vishnu appeared and demanded the armour from Shankhachuda, which he had put on, his body in the guise of a brahmin. Shankhachuda gave his armour to him without any kind of suspicion. Lord Vishnu then went to Shankhachuda's wife in the guise of her husband i.e. Shankhachuda. He destroyed the chastity of Tulsi - Shankhachuda's wife. Shankhachuda derived his power from the chastity of his wife and it vanished the moment, her chastity was destroyed.
1.2.140 Lord Shiva Kills Shankhachuda - Bhadrakali was creating havoc in the army of Shankhachuda. Shankhachuda became very furious and attacked Shiva. Shiva repulsed his assault and attacked him with his Trishul. Shankhachuda who had became powerless was killed instantaneously. The deities were very pleased at the death of Shankhachuda. After worshipping lord Shiva they went back to their respective abodes.
1.2.141 Tulsi Curses Lord Vishnu - After being instructed by goddess Parvati, Lord Vishnu had gone to Tulsi in the guise of her husband- Shankhachuda, so that the breach of Tulsi chastity could help lord Shiva to kill Shankhachuda, who derived his power from Tulsi's chastity and virtuosity.
Initially Tulsi could not recognize lord Vishnu. She was extremely joyous at his arrival. But very soon she was able to realize the real identity of lord Vishnu, who had disguised himself as her husband. She became very angry and cursed lord Vishnu to become a stone. She was crying unconsolably. Lord Vishnu contemplated on Shiva as a result of which he appeared. Lord Shiva blessed Tulsi that she would become the beloved of Lord Vishnu.
Due to Tulsi's curse, Lord Vishnu attained the form of Shaligram which is a stone and because of lord Shiva's blessings. Tulsi leaves started being offered to the Shaligram, in the process of its worship.
That my friend is the complete story of Shankhachuda as per the Shiva Purana.