I can see my tub is chipped a bit on one side of the tub drain and water enters the ceiling from there only while taking shower, need to replace the drain and fill new plumber's puty in but do I have to wait a certain period so that plumber's putty will stay on and not come off cause right now I have just filled the sides of the drain with putty but it never dryes out.

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water enters the ceiling from there only while taking shower,

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This doesn't look like a drain problem. If it were it would be worse if you were taking a bath. This sounds more like the tile grout has shrunk around where the shower stream hits the shower wall. Water hits the tile grout and gets behind the tile and leaks past the shower pan into the ceiling. Caulk the tiles with silicon jel and see if the leak doesn't go away.
The correct way to replace a drain seal is to unscrew the drain from the brass tub shoe and form a ring of putty around the lip. As you tighten the drain lip it will force the excess putty out. Putt will stay soft to form a seal. It isn't supposed to dry out. Good luck, Tom