"A new aussie drama" stand for a new Australian drama?
No, "aussie" would never be offensive. At least, I can't think of any time it would be offensive. It is informal though, so it would sound strange in some formal situations.
I am happy to be called an "aussie" and hear things described as "aussie".
If I said, "I am going down the pub" I might think I am saying "I am going down to the pub" but the "to the" part would be so short it would almost not be there - maybe in very relaxed speech, there would be no trace of it.
Dialects of English are more different when they are spoken than when they are written. You might hear an Australian say something that he/she would never write the same way.