As much as I talk about binary quadratic forms turns out I also have extended concepts pioneered with them to cubic residues.
So I have found the following modular solution.
Given: x3 - Dy3 = F
I can solve modulo N, where N is a cubic residue of D, m is the cubed residue, and r is a residue modulo N, where its modular inverse exists:
x = my + r mod N
3(2my+r)2 = 4Fr-1 - r2 mod N
If you wish to see it used, you can just check:
The ideas I use can be generalized further. But I think I went just to a cubic just to see what it might look like. Result seems very pure to me as in I'm not sure how useful it is.