Yes, Leadbelly apparently did the song twice, in radically different versions.

I haven't heard the one on "Last Sessions" either, though to judge from the Oak "Leadbelly Songbook," it seems to be a fairly straight cover of, and learned directly or indirectly from, the 1937 Lomax field recording made in eastern KY. (Woody Guthrie did a fairly similar cover.)

The much more interesting Leadbelly version was recorded for Musicraft on 17 February 1944, in New York, as part of an 8-song session. It was released on 78 as "In New Orleans" and seems to be so titled on all of the subsequent LP and CD reissues. Lyrics are fragmentary, even by HRS standards, and the song's principal interest lies in its wonderful guitar arrangement, one of the best Leadbelly ever recorded. It's the guitar line that has led to numerous 'Cat threads on the song.

For the record, the lyrics are, again,

In New Orleans (House of the Rising Sun)
(re)composed by Huddie Ledbetter

'Way down in New Orleans,
House called the Rising Sun.
Has been the ruin of a many poor boy
And me, oh God, for one.
'Way down in New Orleans,
House called the Rising Sun.
Has been the ruin of a many poor boy,
And me, oh God, for one.

Go tell my baby sister
Not to do like I have done.
Shun that little ole house in New Orleans
They call the Rising Sun.
Go tell my baby sister
Not to do like I have done.
Shun that little ole house in New Orleans
They call the Rising Sun.

I'm going back to New Orleans
My race is almost run.
I'm going back to spend the rest of my life
Beneath that Rising Sun.
I'm going back to New Orleans
My race is almost run.
I'm going back to spend the rest of my life
Beneath that Rising Sun.