What happens to the human inside the titan when the titan dies? Just to be clear I'm NOT talking about those who where swallowed.  


  We know that titans used to be humans and then somehow they transformed into titans.


When they kill a titan it completely Vaporizes and nothing is left.
So what happens to the human inside them?


  Could it be like when Eren was testing his limits and each time he fused more with the titan body? If so, why could Ymir live in titan form for over 60 years and still transform back into human?
I will try and answer by spoiling as little as possible. 

I have read all manga chapters (including the newest one, 91).


   We have two types of Titans (so far): shifters, and non-shifters. A shifter is a Titan whose human can voluntarily transform into and out of the Titan shape. As far as we know, shifters are only the 9 original Titans. We also know that there is a certain serum that can transform a human into a mindless Titan. This serum was injected to Ymir (the person) and like all other mindless Titans, she wandered around until she accidentally ate a human who was a Titan shifter (Reiner's and Bertlod's friend). As such Ymir transformed into a human, obtaining the power of the so-called "Dancing Titan".


Which brings me to the next spoiler (approximately manga chapter 88 and above):


   Only "The people of Ymir" (Eldians, including Eren and the people inside the walls) can be transformed into Titans. Furthermore, the people inside the walls live on an island called Paradis Island, although far from the shore. Knowing that, the Marley government gathers Eldians on top of a wall next to the shore (but far from the walls), injects them with the serum and throws them on the ground. These Titans wander around eventually finding the walls and creating problems for the people of the walls. (A nice counter strategy from Marley, using a weapon they cannot control). The plot thickens when Marley has managed to convince Eldians to fight for them. These Eldians, such as Reiner, Bertold, Annie, Zeke, each hold one of the 9 original Titans power, and can easily transform from Titan to human to Titan again.


To answer your question

1) If you are talking about Titan shifters

Then you cannot kill a Titan without killing the human inside. The methods used by the Survey Corps are pretty much cutting off their nape. If you cut the nape of a Titan-shifter, then you are left with the human body of that Titan-shifter. If you consume the spinal cord fluid of this human body, you will obtain his/her Titan-shifting power.

2) If you are talking about mindless Titans

Then if you cut off the nape, there is no trace left of the human who was turned into this Titan. As far as it is revealed, we cannot say if the person evaporated, died, or something else. It is assumed that the person inside a mindless Titan dies, when the nape of the Titan is cut off. Hange speculated that the spinal cord and the central nervous system of a mindless Titan functions separately from the body. Maybe that means that the human body simply turns into a mindless Titan, thus leaving no human body inside of such a Titan.

A list of the 9 original Titans is here