How does the Bible differentiate between sickness and sin? Jay Adams is probably the most pre-eminent Christian counselor who is still living. His nouthetic style of counseling, as explored in books such as Competent to Counsel, The Christian Counselor's Handbook and others makes him very well known and respected in the Christian Counseling community.

In his book Solving Marriage Problems, he makes an interesting claim:


  Calling drunkenness a "disease" or a "sickness" rather than a "sin" may seem to thema gracious act, but it is just the opposite. You cannot be more gracious than God. To call drunkenness a "sickness" is to take away hope; there is no pull that will cure such a "disease". But if, as the Bible says, drunkenness is a sin, then there is real hope, because Christ Jesus came not only to free us from the penalty but also from the power of sin. (Solving Marriage Problems, p14)


This strikes me as contrary to worldly ways of thinking to say the least - namely there seems to be an implicit assumption of mine that is challenged here. 

I've long held alcoholism - or at least the genetic predisposition towards it - to be a disease. As a disease of the body, I see this as somehow "less than sin" because, in fact, it is a biological process put in place by God. In the same way that just about every Christian more mature than Fred Phelps doesn't view "blackness" as a sin, I've always assumed that any biological function is simply a fact, and not a "sin."

Interestingly, Adams seems to then fight his own hypothesis, though, saying:


  When, however, there may be an organic reason for any problem, the counselor should always advise the counselee to have a thorough medical evaluation. ... Because organic problems, which result from Adam's sin and the subsequent curse, are not themselves sin, Christian counselors must become thoroughly acquainted with their symptoms and advise counselees accordingly.


But therein lies the rub - Adams has made and I have made an assumption that a thing cannot be both a disease and a sin.  It is definitely my gut feeling, but I don't know how I could biblically prove that point.

So, here's the question - What biblical texts or traditions would bear on the proposition that "Diseases and sins are separate things."  In other words, how could I make either the claim that "If something is a disease, it is not at the same moral level as sin" or "Just because something is a disease, doesn't mean it's a sin."
The full context of Adam's claims are obviously hidden from us here. However, addressing the visible claims from a lay Catholic perspective, I'd say he's simply off the mark with a few premises.

The First Faulty Premise


  To call drunkenness a "sickness" is to take away hope; there is no
  pull that will cure such a "disease".


If secular medicine and psychology aren't enough to convince us that disease is not a hopeless state, we can open any one of the gospels and take a peek at almost any page to find healing upon healing upon healing. And these healings always denote the forgiveness of sin and the purifying power to go forward and "sin no more."

See Matthew 4:24, which gives us a short list of diseases Jesus cures. This items on this list are very important and relevant, mind you -- they're all conditions that directly inhibit our ability to perform God's will.


  His fame spread to all of Syria, and they brought to him all who were
  sick with various diseases and racked with pain, those who were
  possessed, lunatics, and paralytics, and he cured them.


Or refer to Matthew 8:1-17, in which healing the diseased is portrayed again as routine operation for Christ. Even more astounding, note how final verse in the list of healings ends.


  When it was evening, they brought him many who were possessed by
  demons, and he drove out the spirits by a word and cured all the sick,
  to fulfill what had been said by Isaiah the prophet: He took away our
  infirmities and bore our diseases.


One of Christ's missions and trademarks, as explicitly stated in scripture, is to cure the sick, take away our infirmities, and bear our diseases. Thus, I'd submit that hopelessness in the shadow of disease is sin; it rejects Christ's power and mission to heal.

The Second Faulty Premise


  Adams has made and I have made an assumption that a thing cannot be
  both a disease and a sin.


I'm not sure this statement of yours matches Adam's exactly:


  Because organic problems, which result from Adam's sin and the
  subsequent curse, are not themselves sin


From what I can see in your citations, Adam's seems to leave room for the interpretation that the organic roots of sinful behavior are not sinful, but that the resulting actions may still be sinful.

But, giving you the benefit of the doubt, since I'm not familiar with Adam's or his book(s), one passage in particular comes to mind. See Mark 9:42-48.


  42 “Whoever causes one of these little ones who believe [in me] to
  sin, it would be better for him if a great millstone were put around
  his neck and he were thrown into the sea. 43 If your hand causes you
  to sin, cut it off. It is better for you to enter into life maimed
  than with two hands to go into Gehenna, into the unquenchable fire. 44
  45 And if your foot causes you to sin, cut it off. It is better for
  you to enter into life crippled than with two feet to be thrown into
  Gehenna. 47 And if your eye causes you to sin, pluck it out. Better
  for you to enter into the kingdom of God with one eye than with two
  eyes to be thrown into Gehenna, 48 where ‘their worm does not die, and
  the fire is not quenched.’


In this segment, Christ is not necessarily addressing diseases, but He is very clearly addressing biological causes for sin. And in doing so, He makes it very clear that the sins resulting from these biological causes do condemn us. And while we tend to believe He's speaking in hyperbole here, the importance of earnestly trying to cut oneself off from any and all sources of sin, even those ingrained in our own body, is very important to our salvation.

In Summary

We can certainly distinguish between disease (or any biological cause) and the resulting actions, be they sinful or not. However, a biological cause is neither a hopeless condition nor a judgement free condition. For, if the biological causes for sin left us guiltless, Christ would neither warn us against them, nor spend His time healing us and "bearing our diseases." Bearing our diseases is a large part to Christ's mission. So, we're definitely not left hopeless.

Catholic doctrine teaches that our guilt is diminished to the extent our actions are beyond our direct control. However, sinful actions are still sinful even when we are guiltless in performing them. Thus, if we are complacent in our diseases or sins, I would argue I think with the support of Catholic doctrine, that we regain the guilt of those sins!

(And by "guilt" I mean some measure by which we may be condemned.)