The obvious, and very prominent, link between country music and soul in the 1960s was Ray Charles, who was of course born in Georgia ('Georgia On My Mind' being one of his signature tunes). Charles' 1962 album Modern Sounds In Country And Western Music saw him playing in a style with elements of both the 'countrypolitan' sound in country popular in the 1960s, and well, it's Ray Charles: there's soul (the box set of Ray Charles' recordings on Atlantic is called The Birth of Soul, and that's not exaggeration). Modern Sounds covered songs by Hank Williams, Don Gibson, and Fred Rose of Acuff-Rose (hear him do Hank's 'Hey Good Lookin' in a very big band jazz style here). (And if you know of Chris Stapleton, you likely know another musician in the same ilk, Sturgill Simpson, who had a 2014 album called Metamodern Sounds In Country Music, a clear tip of the hat to Charles.) I would also argue that the Texan country singer Willie Nelson (whose debut album And Then I Wrote was also released in 1962) was quite influenced by jazz in his writing, his singing style, and his playing - his series of albums doing covers of tin pan alley tunes starting with Stardust is a giveaway - and that Nelson shows distinct signs of soul in his music (he did sign to Atlantic Records in the 1970s for the albums Shotgun Willie and Phases And Stages, and there is definitely a country-soul vibe on many of those recordings, especially the Shotgun Willie title track).

The other avenue from where country music cross-pollinated with soul in the 1960s (I mean, if we ignore rock'n'roll, which fuses aspects of country music and aspects of 1950s rhythm and blues) is in the post-psychedelic move towards 'back-to-roots' of 1967-1968, personified in The Band's album Music From Big Pink in 1968. While there were rock/country crossovers before The Band (who were largely Canadians, but featuring a white drummer from Arkansas), Music From Big Pink was an incredibly influential record. It also clearly featured a cross-pollination of genres, which particularly emphasised country and soul. The album very clearly has country influences (they cover 'Long Black Veil', originally recorded by country singer Lefty Frizzell) and very clearly has soul influences (The Band's pianist/singer Richard Manuel did a cover of 'Georgia On My Mind' as his signature tune, and they'd also cover Motown tunes live like 'Loving You Is Sweeter Than Ever'. Because of The Band having Bob Dylan's imprimatur (he painted the cover photo, and they had been his backing band on tour), The Band also had a counter-cultural cred; English guitarists like George Harrison and Eric Clapton flew to America and went up to Woodstock (where The Band were based) to meet them, and the eventual location of the Woodstock festival was likely influenced by The Band's association with the town.

But anyway, in 2012, the reissue record label Light In The Attic released a compilation called Country Funk 1969-1975, which was basically a survey of music in the same milieu as The Band, featuring everything from Tony Joe White to Jim Ford to Bobbie Gentry to Bobby Darin. If you're after music that sounds like country musicians doing soul music, this is probably what you're looking for (and speaking of Bobbie Gentry, her big 1967 hit 'Ode To Billy Joe' definitely has aspects both of country and soul, and one that likely influenced The Band's music; Bob Dylan and the Band's 'Clothesline Saga', from 1967, is a straight parody of the song).

First of all, go to YouTube, enter “Rainy Night in Georgia,” and listen while you read the rest of this answer.

To give a good answer to this question, we’ve got to go back a ways. The genres that we today call country music, blues, soul (as well as gospel, bluegrass, old-time, rock’n’roll, Western swing, jazz, & others) all share a common source in the murky past of the American South. That source, though, is a tangled web of African, Celtic, German, English, and many other influences, with musicians of all colors borrowing, sharing, and stealing ideas back and forth so as to make it nearly impossible to distinguish a clear thread. For celerity’s sake, though, we’ll jump from the first meeting of a Scotch-Irish fiddler and a black banjo player in the colonies all the way to 1927, on the banks of the beautiful Holston River in Bristol, Tennessee.

In the summer of that year, the Victor Talking Machine Company sent one of its employees, Ralph Peer, on a trip through the South to record the music of its residents. Various companies had been doing this for a few years, but on this particular trip, in what would be known as the Bristol Sessions, Peer recorded a man named Jimmie Rodgers, and a overall-wearing hillbilly family with the last name of Carter. They would go on to be known, respectively, as the Father and First Family of country music.

Prior to the advent of the recording industry, it is near impossible to neatly divide rural Southern guitar-based music into blues and country music. In its early years, Jimmie Rodgers and other white country musicians had a profound influence on black blues musicians, and vice versa. In fact, a lot of that stuff is damn near indistinguishable. That, however, did not bother the marketing experts. They divided their Southern recordings into “race” and “hillbilly” categories, based on the color of the performer, and targeted them to black or white customers accordingly. This began the rift between country and blues music, which would widen from there on out.

Fortunately for us, though, that rift would be bridged numerous times in the near-century afterwards. Since you asked for country music that sounds like blues or soul, I’ll stick to two examples of that here.

First off, we’ve got honky-tonk. Beginning in the 1940s, and made famous by the Hillbilly Shakespeare himself, Hank Williams, honky-tonk is a style of country music developed for playing in bars, clubs, juke joints, and the eponymous honky-tonks of the South and West. It’s got a thumping, danceable rhythm and commonly features the lyrical themes of booze and broken hearts. The subgenre as a whole (and Hank in particular, who’s primary influence as a child was a black street performer named Tee Tot Payne) owes heavily to the blues, especially to the blues subgenre of boogie-woogie.

Next, we’ve got 60s and 70s-era country soul. Country soul kinda has two sub-genres, swampy white country performers like Tony Joe White and Jerry Reed being influenced by soul and funk, and black performers a la Solomon Burke (a personal favorite of mine) being influenced by white Southern gospel. The runaway absolute best bit of this period is the tour de force that is Ray Charles’ “Modern Sounds in Country and Western Music.”

Listening list:

Early stuff-

Jimmie Rodgers: “In the Jailhouse Now,” “Waiting For a Train.”

Mississippi John Hurt: “Let the Mermaids Flirt With Me.”

Honky-Tonk-

Hank Williams: “Honky-Tonk Blues,” “Move It On Over," "Your Cheatin Heart."

Country Soul-

Ray Charles: “Modern Sounds in Country and Western Music” (yes the whole album, trust me it’s worth it).

Solomon Burke: “Just Out of Reach.”

Jerry Reed: “Amos Moses.”

Modern Honky-Tonk-

Whitey Morgan and the 78’s: “Born, Raised, and Live from Flint."

Modern Blues-Inflected Country-

Tom Buller: “When a Country Boy Gets the Blues.”

Modern Country Soul- (aside from Stapleton)

Brent Cobb- (from Georgia!) all of “Providence Canyon.”

Sturgill Simpson- anything off “A Sailor’s Guide to Earth.”

Also check out Larry Jon Wilson, Tony Joe White, Arthur Alexander, Tompall Glaser, and Delbert McClinton. Finally, everyone should know that Al Green did a version of the best country song ever, "I'm So Lonesome I Could Cry."