A quick note on chords.  The actual name of the Fmaj7* is Fmaj7b5, which looks like this:

e| 0
B| 1
G| 2
D| 3
A| 0
E| x

As played here, you could call them C-Cmaj7-C (without a 3rd/5th for the C and no 3rd 
for the Cmaj7) and F5-Fdim-F5 (no b3rd for the Fdim).

As they're plucked here, it's easier to just make the C/Fmaj7 shape respectively and 
lift up your index finger for the major 7th.  That also allows you to do John Darnielle's 
cool little extra bits.  You can also choose to pluck the G string with a third finger.

Here's the progression.  These progression is fingerpicked.
e|--------------------------------------------------------------------------|
B|---1--0--1----1--0--1----1--0--1----1--0--1-------------------------------|
G|--(0)(0)(0)--(2)(2)(2)--(0)(0)(0)--(0)(0)(0)------------------------------|
D|--------------3--3--3-----------------------------------------------------|
A|---3--3--3---------------3--3--3----3--3--3-------------------------------|
E|--------------------------------------------------------------------------|

Simple enough, right?  Sir Darnielle throws in little flourishes that shouldn't be too 
hard to do.  Just keep making that C-esque shape and pluck around some of the other 
strings too.  If you can figure out a single note run that sounds good, do it.

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
I came to see you up there in intensive care,

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
They had handcuffed you to your bed.

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
There were tubes going into you and out from you,

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
Bright white gauze bandages at your head.

Dm (xx0231@1)        FI (133211@1)am a mole,
                AmSticking (x02210@1)his head above the surface of the earth.
Dm (xx0231@1)        FI (133211@1)am a mole,
                 Am (x02210@1)(here, afirm (x02210@1)downward stroke for every beat)
Sticking his head above the surface of the earth.

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
And they said "Lights out!",

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
And it was lights out.

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
And they gave you your medication.

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
I know what you want and you know what I want:

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
information, information.

Dm (xx0231@1)        FI (133211@1)am a mole,
                AmSticking (x02210@1)his head above the surface of the earth.
Dm (xx0231@1)        FI (133211@1)am a mole,
                 Am (x02210@1) (downward strokes on every beat)
Sticking his head above the surface of the earth.

E-Am x4

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
Out in the desert we'll have no worries,

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
Out in the desert just you and me.

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
I came to see you up there in intensive care,

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C
Out in the desert we'll live carefree.

Dm (xx0231@1)        FI (133211@1)am a mole,
                AmSticking (x02210@1)his head above the surface of the earth.
Dm (xx0231@1)        FI (133211@1)am a mole,
                 AmSticking (x02210@1)his head above the surface of the earth.

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C

C-Cmaj7-C   Fmaj7-F*-Fmaj7   C-Cmaj7-C   C-Cmaj7-C

C-Cmaj7-C-Cmaj7-C-Cmaj7  Fmaj7-F*-Fmaj7-F*-Fmaj7-F*

C.