BELOVED= /bɪlʌvɪd/ someone you love very much (it can also be pronounced /bɪlʌvd/). It may be a noun or an adjective:
Adj- A beloved friend = a friend I love very much
Noun- My beloved, come with me
The surviving old pronunciation here (we usually still pronounce here the E in the –ed ending) is because this word is a bit old fashioned but it is very common in the standard version of the English Bible (King James Bible), which is written in old fashioned English:
“Beloved /bɪlʌvɪd/, let us love one another: for love is of God, and every one that loveth /lʌvɪθ/ is born of God, and knoweth God. He that loveth not, knoweth not God, for God is love” (1 John 4:7-8)
This same verses, in a modern English translations, say:
“Dear friends, let’s love one another, because love is from God, and everyone who loves is born of God and knows God. The person who doesn’t love doesn’t know God, because God is love.” (NET Bible, 2006)

SUGAR /ʃʊgə*/

UR= (col.) You’re. /jɔ:*/
The pronunciation suggested by the spelling UR is /ɜ:*/, but in very colloquial writing we can use U for YOU (same pronunciation).
Also, the verb ARE is pronounced the same as the letter R, so we can use R instead of “are”:
- U R Gr8 = you are great
So we have:
You are = U R = ur
- ur here still = you are here still
The correct order should be “you are still here”, but changing the order makes the STILL more emphatic.

GIMME= (col.) Give me.

I WASN’T FINISHED= I haven’t finished.
- I’m finished = I have finished
- I was finished = I had/have finished.
The verb “to finish” is usually followed by –ING:
- Have you finished cooking?
- I wasn’t finished dreaming about your lips = I haven’t finished dreaming about your lips (= I was dreaming about your kiss, and now that I’m awake I want to continue kissing you because I didn’t have enough)

...YOU WON’T BELIEVE= (also: you wouldn’t believe it). We add this phrase to a statement to express the idea that it is something incredible (so difficult to believe):
- It was so hot there, you won't believe (it)!
- There is so much light in the city, you won't believe =  that you can’t believe it’s real.

BEEN= I’ve been.
In colloquial English we sometimes drop the subject, and sometimes drop the auxiliary verbs, but the context makes clear the subject is “I” (that’s why we can drop it) and the past participle (been) makes clear that there should be a “have” before, so you can drop the subject and the auxiliary and still the sentence makes sense.

NO TIME TO SLEEP= I have no time to sleep.
Again we dropped the subject and the verb “have” (although in this case it is not an auxiliary)

UR= (col.) Your.
As we commented above: You are = U R = ur
But YOU’RE and YOUR are pronounced the same: /jɔ:*/
- You’re Spanish
- Your friend is Spanish
So if we can write UR for YOU’RE, then we can also write UR for the possessive YOUR. And this is what you can find here in the lyrics:
- If ur heart ever fell out = if your heart ever fell out

HEART= /hɑ:*t/

PILLOW= (See picture)

UNLESS= If not.
This negative connector is like IF but negative:
- If you don’t come, I won’t give you this = Unless you come, I won’t give you this.

...UNLESS I’M FALLING FOR YOU = He’s talking about sleep, he doesn’t want to fall asleep, so when he says “I don’t wanna fall” we understand he means “I don’t want to fall asleep”. But then he says “unless I’m falling for you”. To fall for someone means “to fall in love with someone”, so he’s playing with the meaning of FALL here: In the beginning of the sentence we think it means “to fall asleep” but then at the end of the sentence we suddenly discover that he means “to fall in love” (=to start loving someone very much). The problem here is that grammar says both “falls” must mean the same, or the sentence would be incorrect, but meaning says that both “falls” are different, or the sentence wouldn’t make sense in the context of the story in the song. So this sentence is like a surreal bridge connecting to different ideas in one impossible sentence. Nice.