Assalaamu alaykum. There are two widespread stories about the Prophet, sallallaahu alayhi wa sallam, in our locatity which are even mentioned in the school text book. Is there any basis of these stories?
1) Once, a beggar came to the prophet and asked for food. The Prophet did not give him food but bought an axe and gave it to him. Then he went to the forest, cut wood with his axe, earned his livelihood, and thus gave up begging.
2) A man of many sins came to the Prophet and asked for advice. The prophet only told him not to lie. Then the man gave up telling lies and gradually gave up all other sins.

All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.

The first story was reported by Abu Dawood, from Abu Bakr Al-Hanafi, from Anas ibn Maalik, who said:

“A man from the Ansaar came to the Prophet and begged from him. He (the Prophet) asked, ‘Do you not have anything in your house?’ He replied, ‘Yes, a piece of cloth, a part of which we wear and a part of which we spread (on the ground), and a wooden bowl from which we drink water.’ He said, ‘Bring them to me.’ He then brought these articles to him, and he (the Prophet) took them in his hands and asked, ‘Who will buy these?’ A man said, ‘I shall buy them for one dirham.’ He said, ‘Who will offer more than one dirham?’ (He said it twice or thrice.) A man said, ‘I shall buy them for two dirhams.’ He gave the articles to the buyer and took the two dirhams and, giving them to the Ansari, he said, ‘Buy food with one dirham and hand it to your family, and buy an axe with the second dirham and bring it to me.’ He then brought it to him. The Messenger of Allah fixed a handle on it with his own hands and said, ‘Go, gather firewood and sell it, and do not let me see you for a fortnight.’ The man went away and gathered firewood and sold it. When he had earned ten dirhams, he came to him and bought a garment with some of them and food with the others. The Messenger of Allah then said to him, ‘This is better for you than that begging coming as a spot on your face on the Day of Judgment. Begging is suitable only for three people: one who is in severe poverty, one who is seriously in debt, or one who is responsible for paying blood money and finds it difficult to pay.’”

The chain of narrators of this hadeeth is classified by Shaykh al-Albaani as Dha’eef (weak).

As regards the second story, then we have not come across it in the Hadeeth books, but we found it mentioned in some literature books, such as Al-Mahaasin wal-Adh-dhaad authored by Al-Jaahith, and Al-Kaamil fi-l-Lughah wal-Adab authored by Al-Mubarrid.

The text of the story which they reported is as follows:

“A man said to the Prophet ‘I do four things in secret: fornication, theft, drinking alcohol, and lying. Whichever of them you wish that I abandon for you, I will do so, O Messenger of Allah?’ He [the Prophet] said, ‘Do not lie.’ When the man went away, he was about to fornicate, but then he thought: If the Prophet asks me about it, if I deny it, then I will have lied while I promised him not to lie, and if I  confess it, I will be lashed or stoned to death. Then he was about to steal and drink alcohol. He thought the same thing [that if he did not tell the truth, he would break the promise, and if he said the truth, he would be punished]. Then he returned to the Prophet and told him, ‘You have blocked all the doors for me; I have indeed abandoned all of them.’” 

Allah knows best.