Merhabalar! I have made an attempt to translate this beautiful Sıla song, although perhaps a little too loosely. I am guessing that the use of "canım" is twofold here -- to mean "dear" and also to mean "my soul," as in "canım sıkılıyor" -- "my soul is squeezed," or "I am troubled." Also, I´m not completely sure about "nolursun." I´m guessing that this is an ellision of "ne olur." But I don´t know if I understand the "sun" part. ... Links to the music are below. ... As always, I look forward to your explanations. Çok teşekkürler!

Canım Sıkılıyor -- I Am Troubled

Canım bana bırak bu gece gözlerini -- Dear, let me have your eyes tonight.
Aklımı alıp götürme bu gece -- Do not carry away my mind tonight.
Gözlerimde yaş yerine gözlerin olsun -- Let it be your eyes, instead of tears in my eyes.
Nolursun dur gitme -- Please, wait, don´t go.

Canım sıkılıyor canım gözlerinle olmayınca -- I am troubled, dear, when I don´t have your eyes.
Hani koyup gidiyorsun ya beni canım canım sıkılıyor geceleri -- In fact, you are leaving me and going away. Dear, I am troubled in the nights.
Odam buz gibi sensiz uyku tutmuyor gözlerimi -- My room is like ice without you. I cannot sleep.

Canım bana bırak bu gece ellerini -- Dear, let me have your hands tonight.
Aklımı alıp götürme bu gece -- Do not carry away my mind tonight.
Ellerimde kadehler yerine ellerin olsun -- Let it be your hands, instead of a wineglass in my hands.
Nolursun dur gitme -- Please, wait, don´t go.

Canım sıkılıyor canım ellerinle olmayınca -- I am troubled, dear, when I don´t have your eyes.
Hani koyup gidiyorsun ya beni canım canım sıkılıyor geceleri -- In fact, you are leaving me and going away. Dear, I am troubled in the nights.
Odam buz gibi sensiz uyku tutmuyor gözlerimi -- My room is like ice without you. I cannot sleep.

Canım sıkılıyor canım gözlerinle olmayınca -- I am troubled, dear, when I don´t have your eyes.
Canım sıkılıyor canım ellerinle olmayınca -- I am troubled, dear, when I don´t have your hands.
Canım sıkılıyor canım … -- I am troubled, dear ...

YouTube: https://www.youtube.com/watch?v=yz91nvDIOjo

Izlesene: http://www.izlesene.com/video/sila-canim-sikiliyor/8242789

Your (literal) translation is close to OK.

Canım sıkılıyor = I feel bored/depressed/down/etc.

lit. My sould is being squeezed/pressed.

Addition:

n´olur = ne olur=what happens (lit.) means "please"

nolursun = ne olursun = what do you become (lit.) means also "please"

-sun doesn´t add to much to the meaning but it can be used. It would have the same meaning in case -sun is dropped. Maybe speaker thinks it would give some better (more insistent) meaning.

Looks pretty good to me, except I have one small correction for the following line:


Hani koyup gidiyorsun ya beni canım canım sıkılıyor geceleri: You know, when you leave me and go away, I get troubled at nights.


It is true that "canım sıkılıyor" often means "I´m bored", but it is also used in the sense of being worried or troubled, as in a sentence like: "Bu konuya canım sıkılıyor" (This thing is worrying/bothering me, I´m troubled about it). I think troubled is even a better word choice for the theme of this song.

Thanks very much to you both! 

Let me see if I understand about this correction, though. "Hani" is one of those words I find very difficult because it seems to have so many meanings. Here, I guess, "hani" along with "ya" means "you know"? But how do you get the "when" part? To me, this line looks like two present continuous statements run together. I must not be seeing some connection here.

Yes. Hani is kind of a reminder here, she is reminding him of the fact that he leaves her alone. (not that he forgot it but just to bring it up). Some more examples:

Hani Almanya´daki kuzenimden bahsetmiştim ya, yarın beni ziyarete geliyor.

Hani babamın doğum günü var demiştim ya, ona hediye alacağım bugün.

Hani gülüyorsun ya öyle, çok hoşuma gidiyor.

Hani alışveriş yapacağız ya, ekmek almayı unutmayalım.

Actually "ya" alone gives the same meaning. And you can put hani either at the beginning or after ya, it doesn´t matter. As you can see, it is used with all tenses. I think for the sentences formed in the past tense, it can be translated as "do you remember?", but for the other ones "you know" seems to work best.

Anyway, I´m not sure why I put "when" in the translation. It´s not coming from hani, just for a moment I thought it sounded better. But I guess you´re right it´s not really necessary: "You know, you leave me and go away, I get troubled at nights" 

Actually, I think you were right to include "when." In English, I would understand the sentence without the "when," but it wouldn´t sound as smooth. Even though there isn´t a distinct construction for "when" here in Turkish, the sentence needs a "when" or a "well" to feel complete in English. I tried to translate your example sentences, and three of them seem like they need a "well" in English. The "well" just gives a breath and somehow connects the two thoughts. I guess it´s sort of like a verbal ellipsis, or like the brain is jumping from one related but separate thought to another. ... Tekrar teşekkürler!

Hani Almanya´daki kuzenimden bahsetmiştim ya, yarın beni ziyarete geliyor.
You know I had mentioned my cousin in Germany, well, he is coming to visit me tomorrow. 

Hani babamın doğum günü var demiştim ya, ona hediye alacağım bugün.
You know I had said that my father has a birthday, well, I will buy a present for him today. 

Hani gülüyorsun ya öyle, çok hoşuma gidiyor.
You know when you laugh that way, it pleases me a lot. 

Hani alışveriş yapacağız ya, ekmek almayı unutmayalım.
You know we will go shopping, well, let´s not forget to buy bread. 

This is a very crucial piece of information on the Turkish language. There is not 1 word to say ´when´ in Turkish, there are several ways to construct the sentence to mean ´when´. You have to realise ´when´ can mean ´if´ or ´while´ or ´after´ or ´just as´ which all have different translations in Turkish: 

-ince/unca

-dik

-ken

-den sonra

-etc, etc

http://www.turkishlanguage.co.uk/avctime.htm on ´when´

http://www.turkishlanguage.co.uk/iken.htm  on ´-ken´

http://www.turkishlanguage.co.uk/conditional.htm on ´if´

http://www.turkishclass.com/forumTitle_8771 this forum on ´when´

Hope its helpful & good luck with that