Scriptural translation is (like most translation work) not an exact science. The fact is that translations from one language to another can have a lot of difficulties, and can result in confusion about what a text actually means theologically.
For we do not have a high priest who is unable to empathize with our weaknesses, but we have one who has been tempted in every way, just as we are—yet he did not sin.
Hebrews 4:15 (NIV)
This translation of the text adds a few questions we’d need to deal with since it is unlikely that Jesus dealt with specific sins that we would have. He could not be tempted to covet his coworker’s new computer, nor could he be tempted to anti-native Canadian racism, both because the opportunities did not present to him in 1st century Palestine. More to the point, the reference to Jesus not having sin would also mean that the specific forms of sin that come from addiction would likely not be part of Jesus’ experience, because he had never (for example) drunk alcohol to excess with enough regularity to become addicted to it, since with drunkenness being a sin, Jesus Christ would not have ever been drunk.
More difficult would be the questions of issues such as anorexia, bulimia, cutting and other similar behaviours. While these are not listed as “sins” in scripture, they are clearly temptations that people can struggle with and have very negative results in the destruction of a person created in the image of God. Did Jesus when he was human have temptations like these?
I am going to have to say “yes” and “no”. Part of this answer comes from looking at the translation of scripture itself. Here is the likeliest Greek construction of the verse that the NIV translates with “tempted in every way”.
οὐ γὰρ ἔχομεν ἀρχιερέα μὴ δυνάμενον συμπαθῆσαι ταῖς ἀσθενείαις ἡμῶν, πεπειρασμένον δὲ κατὰ πάντα καθʼ ὁμοιότητα χωρὶς ἁμαρτίας.
Hebrews 4:15 (NA28)
The phrase in question is “πεπειρασμένον δὲ κατὰ πάντα καθʼ ὁμοιότητα χωρὶς ἁμαρτίας.”, which is (based on my terrible translating) “tested according to all varied likenesses, yet separated from sin”. The ESV translation team (made up of scholars much smarter and well-trained than I) rendered it thusly:
but one who in every respect has been tempted as we are, yet without sin.
Hebrews 4:15b (ESV)
I think this gets us closer to an answer for whether or not Jesus would have struggled with self-destructive addictions. The text is not saying that Jesus has every individual temptation that we do, but that in every class of sins, he has that experience, and so is able to sympathize with our temptations. I think that includes the temptations that come from physical and psychological dependencies.
Jesus Christ in his earthly ministry would have had all of the weaknesses that flesh is heir to, and thus would have dealt with similar temptations to those created by physical dependency (which is the level of some forms of alcoholism). From the way it feels, I’m told that there’s little difference between a physical and a strong psychological dependency, and we do have an example of Jesus recorded as having dealt with having to resist what would have been, in his context, a sinful dependency, namely his temptation to break his fast in the wilderness in a sinful way.
And Jesus, full of the Holy Spirit, returned from the Jordan and was led by the Spirit in the wilderness for forty days, being tempted by the devil. And he ate nothing during those days. And when they were ended, he was hungry. The devil said to him, “If you are the Son of God, command this stone to become bread.” And Jesus answered him, “It is written, ‘Man shall not live by bread alone.’ ”
Luke 4:1–4 (ESV)
Jesus could have sated his hunger by sinfully turning a stone to bread, but that would have ended the fast he had intended for Godly reasons and done it in a way that would have misused his power as God. Jesus had a physical need for food, and yet was unable to fill that real felt need in a Godly way, and managed to defeat the temptation. In the sense of dealing with a dependency that could only be filled by sinning, Jesus was in fact tempted that way and did not sin.
I think this is actually akin to the experience felt by those dealing with addictions and self-destructive compulsions. It feels like a very real need from the inside (and in some cases may actually be a real need in the case of physical dependency), but to fulfill that need by the easiest felt means would be sinful (whether by cutting, purging, or dangerously refusing to eat, and thus damaging or destroying a person created in the image of God). Jesus does not need to have struggled with the specific sins (drunkenness or self-destruction) to have dealt with the class of sin, and thus to meet the encouragement intended in Hebrews 4.
The upshot is that if you are dealing with an addiction, a psychological compulsion, or any other compulsion that can lead you to sin, the Christian claim is that Jesus has dealt with similar strong temptations, and so is able to sympathize when you come to him to ask for help to deal with the temptations you’re facing, and even to grant mercy and forgiveness when you fall into the sin. He knows the power of sin, even as he has not fallen to it.