As far as the last lines not rhyming, SuperDave, the Stanley Brothers as well as Bill Monroe and Doc Watson sing it""I’ve killed the girl I love you see/Because she would not marry me" in their version.

But then the chorus seems like it can fully rhyme – but can't fully make sense, or vice versa.

This rhymes but doesn't fully make sense (what's beside the river? Not entwining?)
Then only say that you'll be mine
In no other's arms entwine
Down beside where the waters flow
On the banks of the Ohio

This makes sense (their house will be beside the river) but doesn't fully rhyme (and you lose some of the jealous demand for exclusivity)
Then only say that you'll be mine
And our home will happy be
Down beside where the waters flow
On the banks of the Ohio

The best I can think of is to restore the old Pedee (or maybe sneak Mississippi in there or something)
Only say that you'll be mine
In no other’s arms entwine
And then our home will happy be
On the banks of the old Pedee