Does Lord Narayana take incarnation to change era?


"Does Lord Narayana take incarnation to change era?"

Not necessarily. There are few AvatAra-s which are listed for Vishnu, but there may be many others which are not that much famous. IMO, we cannot define when the lord should take an incarnation. Also not everyone of them were born at the end of an era, as there are only 4 era-s.

"The thing I want to know the time of Lord Rama and Lord Krishna death. Means whether they died at end of era OR in between the era"

Yes, End of the era, as you rightly mentioned! This is specific only to RAma & Krishna.
Another AvatAra of ParashurAma was before Shri RAma and is believed to be long lived.

As per MahAbhArata's ShAnti parva:

In the Treta age I shall take birth as [Parashu]Rama in the race of Bhrigu, and exterminate the Kshatriyas who will become proud of their strength and possessions. Towards the close of Treta and the beginning of Dwapara, I shall take birth as Rama, the son of Dasaratha in Iskshaku's royal line. ... Towards the close of the Dwapara and beginning of the Kali ages, I shall again appear in the world taking birth in the city of Mathura for the purpose of slaying Kansa

Similar thing is said by Krishna himself after the death of Duryodhana during Shalya Parva:

Krishna replied, saying, "Thou art always described as bereft of wrath, and righteous-souled and devoted to righteousness! Calm thyself, therefore, and do not give way to wrath! Know that the Kali age is at hand. Remember also the vow made by the son of Pandu! ...

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