I've never used a chanty as a worksong so anything I have to say is pure conjecture, but could the heaves/hauls be just in the second and fourth lines:

Solo (no pulls) Oh, Shenandoah, I'm bound to leave you
a-a-WAY you rolling RIver
Solo..Oh, Shenandoah, I'll not deceive you,
a-WAY, I'm bound to GO, cross the WIDE Missouri.

It doesn't matter if the pulls are irregular in number as long as the crew all know when the pulls come and are working in unison. The metre of the solos wouldn't be that important.

I imagine it as a chanty for some strenuous work that doesn't need doing quickly, so a long drag.