You got the sequence written a bit confusingly so that you don't see the forest from the trees.
I'd split the chords of the verse to two groups like this:
- (1) : Am - E - C
- (2) : Ab - Bb
The Am - E - C part is just regular C or Am stuff, nothing fancy there.
Then the Ab - Bb, that's a "thing" in itself, a commonly used thing. If you know what "licks" are for guitarists, then you could think of this like a lick for arrangers. (You know, arrangers are those people who take a regular simple melody and try to conceal its simplicity by writing lots of weirdo chords under it ... just kidding) A little chord trick, which switches to the key of C minor for a brief moment. Usually in its most common form it would go all the way back to C major like: Ab - Bb - C. That's a funny jazzy'ish little thing you can add to for example the ending of a song that's in C, instead of going straight back to C. For example if the chords are C - F - G - C, and you want to make it a bit more fancy and delay the resolution, you insert the chords Ab - Bb before getting back home to C. Now in this Raw Sugar song, it kind of interrupts this chord lick and instead of ending on C it jumps back to Am instead, starting a new line in the verse.
The scale you could use for melody or solo on those Ab, Bb chords could be Cm natural (3 flats) - which the melody does at least in most places.
The pre-chorus is just the same, only the C, E, Am are in a different order. Again, it uses the trick of switching between the keys of C major and its parallel minor, C minor. You can look at it basically as a key shift of 3 semitones, a very common thing in jazzy music.
The chorus uses the same switch between C minor and C major. Whenever there's C major chord, you're in C major, no sharps/flats, and whenever there's Gm, Bb or Ab, you're in C minor, 3 flats. Or actually ... the Bb is a borderline case.
In a song that's in C major, you can use Gm and Bb as a fancier substitute for a C7 in order to move to an F subdominant in the key of C major.
In the chorus, the Bb, coming after C major, is a halfway in-between "guess what key we're in, is it just a fancy C7, maybe mixolydian feeling ... or are we moving over to Cm". On the Bb, you can play a scale with 1 flat, or 2 flats ... but when the Ab comes, it's a dead giveaway that we're (temporarily) in Cm and you should use a scale with three flats. If you left the Ab out and jumped from Bb to Gm, then it would leave the question open, you could interpret it as being in C mixolydian or "fancy C7". But now that there's the Ab, the Gm is seen as being in Cm.
If you really want to learn this, you try it on songs and mess around with the harmony. Take a song book and find a song that's in C major and ends on a C major. Right before the ending C major, insert Ab and Bb! And try playing a Bb or Gm in place of any C7 you see in a song. You only learn by trial and error in practice. And play the scales I mentioned over this song. You don't really learn much with theory. 99% practice, 1% theory. You already used up your 1% for about a week, so go and spend the 99% practicing... well OK, just trying to emphasize the importance of practicing, trial and error, in this modern world which values theoretical thinking too much, in my opinion.