How does the Bible differentiate between sickness and sin?

Disease and sins are not separated.
The man having infirmity 38 years heard from God:

sin no more, lest a worse thing come unto thee. (John 5:14)

Disease comes always as consequences of sins, because:

God saw every thing that he had made, and, behold, it was very good (Genesis 1:31)
by one man sin entered into the world, and death by sin (Romans 5:12)

Sometimes, the connection between sins and disease are not obvious because the sins are inherited from ancestors:

your children shall wander in the wilderness forty years, and bear your who*edoms (Numbers 14:33) For all have sinned, and come short of the glory of God (Romans 3:23)

because the human nature can carry across generations, from one person to another, not only the sins (from Adam), also the virtutes, also the knowledges, also the blessings and cure(from Lord Jesus Christ).

our old man is crucified with him, that the body of sin might be destroyed, that henceforth we should not serve sin. For he that is dead is freed from sin. (Romans 6:6-7)

The root of sins are localizable in the human souls, and the diseases in the human bodies.
This is why, medical science cannot address sins, because the soul is not subject of science, or it uses a wrong definition of the soul (in psychology or psychiatry). At most it can record patological manifestations of the sin at the body level, and make statistics, but cannot administer real cures.

Only a true christian can have a complete view of human anatomy, and can address the true cause of the diseases: the sin.

Is the problem in the deffinition of Sin? As an evangelical, I'm going to give what I think is an evangelical view point.

Sin is not the act itself, it's the breaking of relationship with God - or to put it another way it is us telling God that we know better than Him and will do it our own way. We live in a sinful world - it is broken and not perfect and a consequence of that our bodies are not perfect (not because of any particular sin, but because of the sinful world).

This however doesn't excuse us from our sin. We are accountable for our words and actions (E.g. Matthew 12:36-37, Romans 14:12). Although there might be a biological reason that we might be pre-disposed to being an alchoholic, doesn't mean that we have to drink. In a similar way, our biology/environment/upbringing/education/experiances might mean that we have a violent nature - this doesn't excuse us from acting on that nature and hurting or killing another human.

Again, there is an argument particularly in the western world today that being gay is not a choice - you are born that way. From a hard line conservative evangelical point of view, even if it is true that you were born that way, it doesn't excuse you from dissobaying God and being in a h*m*sexual relationship.

Basically you are born sinful, even before you take your first breath, you are sinful. BUT this does not excuse you from your sin, you are accountable for all of your sin.

This is not a full answer, but one place to look is John 9:1-3.

Jesus seems to say that the man he went on to heal of blindness was not blind as a result of either his own sin or his parents'.

I would personally interpret this to mean the disease was not the result of any specific sin at all. What is the cause or purpose of suffering, of course, is "the" big question, so I won't try to answer that one!

With regard to your example of alcoholism, I think that your quoted author has (perhaps intentionally) misrepresented the position of secular medics. The purpose of a doctor diagnosing alcoholism as an illness is to give it a label, in order to simplify the process of giving help. "If patient X is diagnosed with disease Y, then the textbook says give treatment Z."

While many people in society will see a technical name or a medical diagnosis as a way of saying "it's not your fault, it's your disease", a doctor's professional role is not to assign blame, and not to absolve it either. We should all see the label in that way, and then act according our love for our neighbour.

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.)