Is lying in this case considered permissible lying, and is the solemn oath considered a "ghamūs" (oath that dips the taker in sin) oath, and what are the consequences of that?

The act you performed by sending charity to the man and concealing it is a good deed. However, you erred by swearing a false oath, and repentance to Allah is obligatory. You could have used tawriya, which is speaking words that carry two meanings—the speaker intends one, and the listener understands the other. This is permissible when there is a need or a legitimate religious interest. Al-Nawawi says: "If a preponderant legitimate interest calls for it, outweighing deceiving the مخاطَب (the addressed person), or if there is a need from which there is no escape except through lying: then there is no harm in ta'reedh (euphemism/indirect speech)." Some scholars have gone further, permitting lying and swearing falsely for a preponderant interest, citing the hadith of Umm Kulthum bint Uqbah, that the Prophet (peace be upon him) said: "The liar is not he who reconciles between people, narrating good or saying good... And I have not heard him permit anything of what people call lies except in three cases: reconciliation between people, war, and a man's conversation with his wife, and a woman's conversation with her husband." However, tawriya remains an escape from lying.