There are many stories but one story i know is same as what you have shared in the question.
One day, Bhagwan(Lord) Shiva and Parwati went on a tour. They arrived at a temple in a place called Amrawati. Goddess Parwati asked Mahadev to play a game of Chausar (a dice game). When they were about to play the game, a brahman came and started preparing for aarti. Goddess Parwati asked the brahman to forecast the name of the winner. The Brahman said, "Lord Shiva will win the game".
Now at this point we have multiple stories.
Story 1: Parwati looses the game and she thought it was brahmans prediction that made her loose the game and as a result she cursed the brahman with a disease called leprosy.
Story 2: Parwati wins the game and because of the lie told by the brahman she cursed the brahman with a disease called leprosy.
One day an Apsara of Indra lok found the brahman. The apsara told brahman to worship lord shiva for 16 consecutive mondays. After worshipping for 16 days, the brahman got rid of the diesease.
Goddess Parwati then asked the brahman about the transformation. Brahman tells her about the sorah sombaar varta and its procedure.
The son of goddess parwati (kartikey) used to be very disobedient, so goddess parwati thought of doing the varta for her son. She did the varta and as a result the god kartikey became obedient.
God Kartikey then asked goddess parwati about the transformation. Goddess parwati told him about the varta.
A friend of lord kartikey was not getting married. So, lord kartikey advised his friend to do a sorah somvaar varta. After his friend's varta got finished, his friend heard an announcement. A king of a palace wanted to find a groom for his daughter. The groom shall marry his daughter if he gets a garland from an elephant. Lord Kartikey friend did get a garland and as a result got married to the princess.
Kartikey's friend tells the his wife about the way he got her. After knowing the procedure, princess thought of getting a good son. She does the varta and after some time gets a wonderful son.
When the son grew old, he asked her mother about he got born. (Sorry don't know how to put this but i think the son wants to know how an ordinary man (his father) married a princess (his mother)).
Now, the son did a sorah somvaar varta so that he could be a king of a kingdom. He did the varta and after some time a proposal of marriage comes in his doors. The son marries a princess of some other kingdow. Later, the son's father in law dies and since his father in law had no son he then becomes the king of that other kingdom.
Now the new king wanted to visit Lord shiva's temple. So he tells his wife to prepare for materials for puja and asks her to visit the temple. The wife disrespects the puja and decides not to go any where, gives the puja materials to their servant. Seeing this, the king got unhappy and tells his wife to leave the royal palace.
The wife (rather queen) left the royal palace and as time passed she got poor and weak. The queen asked for help from people but every time some kind of mishap would occur and she had to suffer.
One day the queen decided to take rest under a tree. As soon as she lied under the tree the leaves started to shed. All the leaves were out of the branches. Because of this the whole forest started to become naked. No green trees and as a result no food for animals. One day a priest came by and met the queen. Looking at the the poor lady, The priest asked her to be in his ashram.
The queen went to the ashram and all the mishaps started to happen. The queen then told her about the reason why all these bad things were happening. So, the priest advised her to do the same things that was done by her husband. She did the sorah somvaar varta and in the 17th day her husband (the king) started to miss his queen. He started searching for her and finally found her.
Conclusion: Anyone who worship lord shiva on Monday gets good health, wealth and desires get fulfilled.
OM NAMAH SHIVAAY, HAR HAR MAHADEV.
I got a message that wants me to give a source, that's why putting this update.
I read a book called Hindu Vrat Kathayen published by Diamond Books. It's a small book but contains nice stories. You can read the chapter "Sixteen Monday Vrat Katha". The only different thing in the book and in my answer is the very first sentence.
In the book, it says Bhagwan Shiva and Bhagwan parwati wanted to marry in a mortal world and that's why they visited Amrawati.
You can also hear the story that was shown in IBN7 channel. Well, it's actually a daily horoscope show where the story of somvaar varta was told (and also it's in Hindi). I have presented the youtube link below.
Found some web links, don't know if it can be used as reference but people can still read it:
NOTE: These are stories so you might not find about it in our holy books.