I don't know the exact thing he does in the verses, but I just do something similar and it sounds fine:
E Step1 Step2 E
e|-------------------------------|
B|-------------------------------|
G|--1------1------2------1-------|
D|--2------2------2------2-------|
A|--2------2------2------2-------|
E|---------2------2--------------|
I use my thumb in this but you can also re-finger your E using your pinky finger.
(Listen to the song for timing ofc)
[Verse]
E
Come sick and sore
E
Come black with sin
E
Come rich and poor
E
I'll let you in
E
Come defend them
E
Come learn to live
E
You'll see a change
E
I promise it
[Chorus]
C#m B A E
Down at the waters edge
C#m B A E
Swimming in the shallows
C#m B A E
We're all friends here
G# A E
And that's what matters
G# A E
That's what matters
[Verse]
E
Come young and old
E
Come sunk with shame
E
Come back, come home
E
Don't ever leave
[Chorus]
C#m B A E
Down at the waters edge
C#m B A E
Swimming in the shallows
C#m B A E
We're all friends here
G# A E
And that's what matters
G# A E
That's what matters
[Instrumental]
C#m G#
C#m G#
B A
C#m G#
C#m G#
B A
E
[Verse]
E
Come sick and sore
E
Come black with sin
E
Come rich and poor
E
I'll let you in
[Chorus]
C#m B A E
Down at the waters edge
C#m B A E
Swimming in the shallows
C#m B A E
We're all friends here
G# A E
And that's what matters
G# A E
That's what matters