I took some time to think hard on this one before answering, and also to double-check some Scripture.
When it comes to things like addiction - alcoholism, drug addiction, sexual addiction, and the like, the Bible never refers to them as "sickness". It refers to certain acts as sinful, or foolish. Drunkenness, for example, is never called a sin outright, but it is called "foolishness". Sexual sin is, of course, called sin (when it comes to a sexual addiction).
Modern psychology does recognized these types of things as illnesses. Scripture wasn't written with the concepts of modern psychology in mind. I'm not saying that these things aren't sicknesses or diseases of a sort, but nowhere in Scripture will you find such a claim.
With that in mind, from a purely Scriptural perspective, with no room for interpretation, Scriptures don't ever state directly that these things are illnesses.
But there are a lot of things that Scripture doesn't say that are still true. Silence or lack of Scriptural statement doesn't necessarily mean something can't be true. Perhaps a silly example: Scripture doesn't say that we will one day use computers to communicate - but clearly we do.
It also doesn't say that sickness (the more general form) is caused by bacterial or viral infection, improper nutrition, or the like. It does give practical advice in how to combat sickness.
Likewise, it doesn't say that addiction is a sin, but it does say how to combat it - don't do it. Avoid drunkenness, flee sexual immorality, and it does address drug use - the original word is Pharmakeia, which is unfortunately translated as sorcery these days, but it's the same word from which "Pharmacy" comes. Again, the Biblical advice - don't do it. It's sin and avoid it.
But as to whether addiction is a disease... Scripture doesn't say anytyhing, but since a lack of a statement indicating something is true doesn't necessarily mean it's false, all we can say with certainty is "Scripture doesn't say."
Therefore, this is one of those questions that can't be answered from a purely Scriptural perspective, as asked. (That's my answer)
And it's likely to be continued to be debated until we meet God and can ask Him ourselves.
Going beyond my answer, and just offering some thoughts, that are worth considering, in support for Jay Adam's statement...
I read an article several months back (and can't for the life of me find it to link to it) that talks about "the illusion of free will". Most of the article is not on-topic for this question, but there was an interesting statement.
(Paraphrasing from memory) While modern science indicates that free will is an illusion, the illusion of free will has an
evolutionary advantage in that acting as if we have free will helps us
The article goes on to give some examples. Crossing the street was one of them. Someone who believes in ultimate predestination, with the attitude "What's going to happen is what's meant to happen" may just step out into the street without looking, and therefore get hit by a car. Someone who believes their actions are their choice and that they can control the future( to some point) by predicting likely outcomes and acting accordingly will know that looking both ways first is an opportunity to affect future events by deciding whether or not to step out into traffic.
Applied to the question, the thought process used by Jay Adams is somewhat similar. And I've seen it many times myself. Telling someone that addiction is a disease, and not their problem often reinforces the idea that it's not their fault, and that it's out of their control. So rather than working toward a solution, they fall into the trap of self-imposed helplessness. They see it as something external to themselves, and since it's a disease, not something they've chosen to do, they can justify that next drink, or that next hit.
Of course, that's not universal. Plenty of people with a disease fight like crazy to beat it. And I'm not saying that these things aren't diseases of the sort. But I've seen personally people that have been addicts for years, that believed it was a disease that were transformed both by the saving power of Christ and by changing their mindset (the definition of repentance) and viewing their addiction as sin. This was the first step in deciding to beat it. This is the basis of the Reformer's Unanimous program, which is a very successful addiction recovery program based on such principles.
Again, it's not always that way, and I'm sure there are plenty of people that are more effectively helped with the "Addiction as a disease" modern psychology approach. I am absolutely not stating that it's not a disease, or that it's 100% counter-productive to label it as such.
I'm also not saying that those who treat it as a disease aren't doing the best they can, that they don't mean well, that they're not often effective. I'm not saying any of that.
I'm just expanding on the reasoning and personal examples of support for Jay's statement.