I've always taken "my race is almost run" to mean the speaker's days are numbered, he/she is not long for this world, is on his/her last legs, is ready to cash in his/her chips, etc., the implication being that the wages of sin are death; i.e., dissipation has ruined his/her health.

As for "going back to New Orleans to wear that ball and chain", I always understood it to indicate that the dissipated and destructive life within the confines of THOTRS had some sort of irresistable claim on the speaker, whether for psychological reasons, drug addiction, force of circumstance, or some combination of the preceding. What always gave the song its particular pathos, for me, was that awareness on the part of the speaker of his/her self-destructive course and his/her seeming inablility to alter it.