I've heard things from lineraly rising potential to linearly rising force. Since the change is over distance.... which is it? If dF/dx is constant, then energy, being the product of force times distance, would increase at twice the exponential. That would mean that potential does not rise linearly with distance, assuming that there is only one correct definition of potential (which I think probably isn't the case).

I just found a Wikipedia article which makes this apparent contradiction more visible:

"The force between quarks increases as the distance between them increases, so no quarks can be found individually."

That statement is just a careless mistake, typical of Wikipedia.
The rest of the article is reasonable, but the statement
"there is no analytic proof that quantum chromodynamics should be confining," has to be kept in mind.

I should have mentioned that linear (for the potential) confinement is not the only guess. Many people use a bag model or SHO (force increasing llinearly) without much justification other than ease of calculation.
Most quark model predictions are not sensitive to the confinement mechanism, or even to confinement at all, so it is still an open question.

Keep in mind we are dealing with a many body calculation, including nontrivial radiative corrections. The form of the potential is thus largely intractable, outside simple analysis of basic diagrams, accessible only at certain regimes. Worse, perturbation series will completely break down at most energies of interest, and so we can only hope to calculate short range potentials b/c of asymptotic freedom.

Already you can see terms that give contributions both to repulsive as well as attractive behaviour (for instance quarks in the color singlet configuration attract strongly, whereas when they are in the octet its repulsive). Naively you can calculate this and see you will get short range potential not unlike the coulomb potential.

You can model long range potentials however you want, but experiment gives results that are too coarse to probe exact details on its structure yet (that will need to come from theory dealing with the many body problem, as well as significant progress in lattice qcd)