Saw some posts placing the capo on the 6th Fret. I think  Eric actually plays 1/2 
step down as I do. 
It is a standard G, D, A, Bm "chord shaped" progression. If your in standard "E" 
tuning you can still play these chords. 
It will just be in a different key than his track. Trick is finding your "Key" or 
the pitch you sound best in. 
Chord progression 1 G-D-A-Bm
Chord progression 2 D-A-E-Fm
Chord progression 3 F-C-G-Am
There's different ways to do it, but you get the idea. "FIND YOUR KEY"
I have very minimal "Music Theory" or "Circle of 5th's" training or knowledge.
But just learning this much has helped me expand drastically in my live shows.
So if you want to play in his tuning, Bring your guitar down 1/2 step and play these chord shapes.
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Eric Church - Sinners Like me 
Intro  G-D-A  G-D-A ...pause
Acoustic Strum - G-D-A-D-G-D-A-D

G        D            A           D
I was fifteen when my Daddy's old man
          G       D                A
caught me halfway through my first beer
G             D         A              Bm --hang
He laughed so hard when my face turned green
            G           D            A              D --hang
He said you come from a long line of sinners a like me

    G         D             A           D
Now me and my brother go to see him sometimes
             G            D      A
but he don't have much to say anymore
      G          D                A             Bm
So we sit on his headstone with a fifth of Jack D
G           D            A            D
Here's to a long line of sinners like me
      G     D         A     Bm
La de da de da La de da da de de
  G           D            A            D
I come from a long line of sinners like me

   G           D               A           D
My Momma had a soft spot for a hell raisin boy
       G       D              A
So she had two more just like him
G           D              A     Bm
It takes an angel to raise a family
     G            D            A            D
that comes from a long line of sinners like me

         G        D             A          D
Well now maybe he knows one day ill settle down
            G          D          A 
and give my Dad a grandson of his own
    G               D                A             Bm
And when the Doctor smacks him he'll prolly take a swing
         G            D            A            D
cause he comes from a long line of sinners like me

      G     D         A     Bm
La de da de da La de da da de de
  G           D            A            D
I come from a long line of sinners like me

Instrumental--> Bm-A-G--Bm-A-G--Bm-A-G--A

       G     D                A         D
On the day I die I know where i'm gonna go
       G         D                A
Me and Jesus got that part worked out
     G           D        A          Bm
I'll wait at the gate til his face I see
    G          D            A            D
and stand in a long line of sinners like me
  G          D            A            D ---let ring & hang
I stand in a long line of sinners like me

      G     D         A     Bm
La de da de da La de da da de de
  G           D            A            D
I come from a long line of sinners like me
      G     D         A     Bm
La de da de da La de da da de de
G           D            A            D
Here's to a long line of sinners like me
  G           D            A            D --->Ring out...
I come from a long line of sinners like me

Thanks for checking it out folks, I hope it helps!