Note: Hopefully needless to say, but there will be quite a few uses of racial slurs in what follows

The 'popular' songs which are passed down to us, for the most part, lack explicit racism. Songs like "Maryland, My Maryland" and "Bonny Blue Flag" stir with martial energy, and for the most part, continue to perpetuate the Lost Cause ideology that permeated Southern identity, and American memory of the war. The latter song, considered to be something akin to the Confederate National Anthem, opens with nothing short of a full throated paean to the ideology of states rights without implication of just what right that might be:

We are a band of brothers, and native to the soil,
Fighting for our liberty with treasure, blood, and toil;
And when our rights were threatened, the cry rose near and far,
Hurrah! for the Bonnie Blue Flag, that bears a single star.

There is a reason songs like this now populate the proverbial 'Civil War Songbook', it isn't merely because they lack overt racism, but because they also evoke a specific image about the antebellum South, and the traitorous cause which glorified and valorized it. But these are all incredibly sanitized works, and songs of the time certainly could use lyrics and sentiments which are deeply, and unsettlingly racist. This is true of both the South and the North, and although your question speaks to the South originally, I'll be addressing both as I think the contrast in how it manifested itself is especially telling and important in understanding the competing discourses between the two, as in many ways the songs we find from the North are the ones which sound more openly racist, while the South it is more baked into the undercurrent.

And on that note, to be sure, there can also be racial undercurrents in even the 'old standards'. "Dixie" after all started as a minstrel tune - a style far more popular in the North, where the song originated, than the South - and there was even some controversy in the south about whether it was appropriate to adopt such a song, which was originally stylized as if sung by enslaved black people (the original lyrics stylized "I Wish I was in de land ob cotton"), but in the end that was deemed to add to the appeal as it spoke to the devotion of the enslaved to their southern home, "assur[ing] them that African Americans were happy as slaves and loyal to the new republic". But "Dixie" too is quite sanitized now, and what racial underpinnings it did carry, unlike the song implies, quite forgotten, and in any case, it hardly stands as an example of just how racist lyrics of the time might get.

One thing that is exceedingly common with Civil War era music, and which I've previously written on here and here was how many different versions of a song might exist. The addition of new stanzas, the replacement of existing ones, or the wholesale rewriting of songs, often across sides. The reasons were all quite varied of course, and in war time as one might expect many of those were just the product of bored soldiers improvising as they went along.

In the South, this often was specifically done in a way to keep the old tunes while reshaping a song for Southern nationalist sentiments, such as one attempt made to salvage the venerable "Yankee Doodle" into a new tune dubbed "Farewell to Yankee Doodle,” which included multiple stanzas on race, both to directly defend slavery as a benefit to the enslaved, as well as to decry the hypocrisy of Northern racial attitudes:

Doodle’s morbid conscience strains, 
With Puritanic vigor
To loose the only friendly chains 
That ever bound the n----r.

Yet, Doodle knows as well as I,
That when he’s come and freed ’em. 
He’d see a million n----rs die, 
Before he’s helped to feed ’em.

Northern rewrites in turn sometimes might choose to poke fun at Southerners, and while not written in justification in the same way as the above, one such rewrite of "The Bonny Blue Flag" spared little in terms of its use of language, titled "N----r on the Brain". It called out the Southern soldiers for their obsession with slavery and suggested bullets, in the lyrics dubbed "Grant’s anti-n----r pills" as the solution to their problem.

And while that example is racist in of itself, other lyrics found in the North could be a good deal more pointed in their racism. While strictly speaking not Confederate, complementary examples abound coming from Copperheads in the North who opposed the war in various degrees. A popular song in 1863, written as a response to the Emancipation Proclamation, was an explicit rejoinder about the elevation of abolition as an explicit war aim, lamenting how now the war was a:

Fight for the n----r, the sweet scented n----r, 
The woolly-headed n----r, and the Abolition Bill.

Another song of the period, titled "The Negro Man Rising" similarly pushed back against African-American rights by attempting to stoke the fears of whites about what was to come if equality was granted:

Among the belles, the n----rs new styles and airs assume; 
They’ll gain the heart of Beauty, for Beauty loves perfume. 
This, sad experience taught me; to ask my love I went, 
Said she, “I prefer a man of ‘African descent.’”

Similarly, a very popular minstrel tune by the songsmith William Shakespeare Hays entitled “N----r Will Be N----r” was practically pro-Confederate in sentiment in how it portrayed its subject of black soldiers in the USCT as useless and cowardly on the battlefield, reading:

Dey sent us out upon a Scout—an’ we each had a gun,
De Rebels made a dashin’ raid, you ought to see us run, 
An’ I’m satisfied de N---a’s would rather run dan fight, 
Kase a N---a will be N---a, you kin neber make him white.

It is also worth noting that even for those in support of Northern war aims, songs which were intended to be positive could still feel deeply racist to modern ears, slurs aside, due to the performance in minstrel style, with the heavily stylized words meant to evoke an African-American voice, even though the performers would simply be white men in blackface. One such example is "The Song of the Contraband" which was sung from the perspective of a formerly enslaved man who had escaped across to Federal lines and freedom, and singing:

I cum from old Virginny, an’ dey call me contraband, 
To hot for dis yer n----r in Old Dixie land.
Now I’se down upon de South, but I didn’t use to be, 
’Kase dey want to bust de Union: dat’ll nebber suit me.

I would also though make note of a different use of the n-word, in a marching song of the 54th Massachusetts, the most famous of the black units to fight in the war, whose inclusion of the word is a clear commentary on Northern racial attitudes, but from a black perspective:

McClellan went to Richmond with two hundred thousand brave; 
He said “keep back the n----rs,” and the Union he would save. 
Little Mac he had his way, still the Union is in tears,
*Now* they call for the help of the colored volunteers.

These examples of northern songs provide something of a contrast to those coming from the southerners. In the case of both "Dixie" and "Farewell to Yankee Doodle” the mentions of enslaved subjects are less about pointed racism than they are about justifying the cause of slavery. Arguably then, even something like "The Bonny Blue Flag" fits in there too as we all know just what they mean by "when our rights were threatened". The North is the one which ends up being much more concerned with race, whether the casual racism of the soldiers songs, or the abject racism of Northern Democrats who weren't interested in fighting for the "n----r".

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