Is it permissible for me to advise people by quoting Prophetic hadiths with a similar wording, or by mentioning part of them or their meaning, and does this fall under the category of lying about the Prophet (peace be upon him)?

Calling people to Allah is one of the noblest and most sublime deeds. It was the mission of the prophets and messengers and those who followed them. Allah praises those who tread this path. It is a condition for the caller (Da'i) to have insight into what he is calling to, but it is not a condition for him to be knowledgeable of the entire religion, based on the saying of the Prophet, peace and blessings be upon him: "Convey from me, even if it is a single verse." It is permissible to narrate a Hadith by its meaning for someone who is knowledgeable of the language and is confident that the meaning will not be altered, provided that he understands its meaning linguistically and the intent of the one from whom it is narrated, and that there is a necessity for it, such as forgetting the exact wording while preserving the meaning, and that the wording is not an act of worship in itself, like the remembrances (adhkar).