Today, I stayed up all night last night and have been awake over around clocksince.

I'm almost falling asleep soso tired that I fainted for a while while I was sitting.

Today, I stayed up all night and have been awake over around the clock.

I understand what you're trying to say here, but the phrase "around the clock" doesn't work well with the verb "awake". It's meant more for activity verbs like "working around the clock" "studying around the clock".

Now, I can't think straight.

No comma needed

I'm almost falling asleep so that I fainted, I even nodded off for a while while I was sitting.

Fainting is a bit too strong of a verb here. "Nodding off" is a better suited, where someone is accidentally sleeping for a few seconds-minutes

I have to go to bed soon, and I bet I will sleep over twelve hours.

No need for "and" here.