Busy meaning in Urdu is مصروف - masroof. What is the synonym of Busy Synonym of Busy is Active, assiduous, Brisk, busybodied, diligent, employed, Energetic, engaged, engrossed, Full, fussy, haunted, indiscreet, industrious, interfering, in use, Lively, meddlesome, meddling, no...
It seems to me that you can use "busy with" anywhere where the meaning is not that you are in something. For example, "He's busy in the garage." Your "busy in" examples are atypical, and sound strange to me. I would simply leave out the "in" before the gerunds. "...the...