Johnny Cash is the sort of Paragon people usually point to, and there is a point to that, he is pretty good in that he invented the basic rhythm and system for rock as we know it. If he or some band with a similar sound hadn't stepped in at some point, rock and roll would simply have been increasingly electric and metalized jazz, Classical, and blues. Now, I love me some Bluesy AC/DC and classical Prog rock as much as the next guy, but it's nice to have that third little influence that Johnny brought to the table. My personal favorite of his is his cover of "Ghost Riders in the Sky" but honestly you've probably been pointed his way at some point or another already, assuming this isn't the first time you said you don't like country music, so I'll spare you the arguments in this post and move on to other artists.
Jerry Reed is a little bit corny but charming for it, being a 70s man and all. If you're a Funk Enthusiast like myself, his guitar is almost Hendrix-like and just entrancing to listen to. The reason people associate that stock wavy electric guitar lick with something like a Louisiana swamp is because of this man right here. Here's him doing an old blues/folk song to break you into the genre slowly.
There are plenty of songs in the old cowboy genre of music that don't sound like the typical shit you'you're probably thinking of when you say you don't like country music. In fact, stuff like "Tumbling Tumbleweeds" sounds a lot more like relaxing Hawaiian music than country music until you hear the lyrics. I'm too lazy to link all of these on my phone, but everything from "Back in the Saddle again" to "Big Iron" to "Jingle Jangle Jingle"* is more like a catchy little 50s ballad than a Beer Truck Anthem. (*Maybe take the jazzy covers of Jingle Jangle Jingle over the Gene Autry version.)
If you're a bit of a metalhead, try Clutch's distinctive southern flavor of psychedelic madness.
And last but certainly not least, while it's fucking impossible to confine someone like Tom Waits to a Genre, some of his stuff does have a very country/folksy influence if you go looking for it. Notable example being "Gun Street Girl".
So there you go, some Country stuff that isn’t bad, and songs bery much beyond the veneer of hypercommercialized garbage that stadium/country-pop has become. Although if you don't like Clutch, Jerry Reed, or Tom Waits, I'm afraid you're simply beyond all help.