This Reverso entry for camé equates that word with the two you mention for drug addict (drogué and toxico), along with toxicomane and accro.
I’d classify camé, toxico, and accro as slang terms; drogué as being more formal; and toxicomane as perhaps even a medical term.

For “addictive drug,” the same entry lists came (with no accent and short for camelote) as a familiar term for drogue (drugs) generally (near the bottom) and higher on the page the general reference to drugs (drogue and stupéfiant) is repeated as well as a reference to the specific drug of cocaine.

Good evening.

Here’s my answer for your 3 expressions:
-For an addictive drug, you just need to say “drogue” in french. You won’t have to use that word for medicines, since talking about “drogue” also includes smooth and heavy addictive drugs.
-For a drug addict, I consider the word “toxicomane” as the best solution, since it deals with heavy and smoother drug addict.
For a slang expression, you can talk about “toxico” or “drogué”, of course. We can also use these words, such as “un(e) addicte” ou “un(e) accro”.

For example, “c’est un accro au cannabis”, “une addicte de/à l’héroïne”, even if both of these words are more often used for dangerous passions.

There are many more words coming from colloquial and slang french to talk about addiction, but I can’t give you all of them right now.^^

Hope it will help you.

Je suggère:

• addictive drug: drogue, stupéfiant, narcotique

• drug addict: drogué, toxicomane

• slang for drug addict: camé, junky, défoncé, shooté

Ces termes ne concernent que les substances illégales, c’est-à-dire pas l’alcool, la nicotine ou les médicaments psychotropes.

On drogue and addiction

As has been already mentioned by others, the term drogue carries the assumption that the substance is addictive, and that explains why "addictive drug" is not often used in French.

There is a reason for that: the meaning of French drogue is far more restrictive than the English "drug". In English the word has two meanings:

1. A substance used in the diagnosis, treatment, or prevention of a disease or as a component of a medication.

2. A chemical substance, such as a narcotic or hallucinogen, that affects the central nervous system, causing changes in behavior and often addiction.

(Source: American Heritage)

In French, then first meaning is largely obsolete. Hence, when a person says drogue in general conversation, it is generally obvious that they are talking about a mind-altering substance:

Produit stupéfiant ou hallucinogène (comme la marijuana, la mescaline, le L.S.D., le haschisch, l’héroïne, l’opium, la cocaïne) dont l’usage peut conduire à l’intoxication, l’accoutumance et la toxicomanie.

(Source: TLFi)

Admittedly, the "addiction" (French equivalent is dépendance; while accoutumance is the adaptation of the body to a dosage) is not medically considered an automatic consequence of drogue; but it is a high risk. Hence general French usage, every drogue is considered addictive (the -ive suffix expresses the risk). By contrast, in English it might be useful to specify that, only to clarify that we are not talking about usual medicine.

So you might give up translating the word "addictive", because it is already included. Or else, if you really meant a "(strongly) addictive drug" a periphrase would be more usual:

drogue provoquant une forte dépendance

Other terms

Drug addict is indeed, in formal language:

toxicomane

The must common word (not slang and not offensive):

drogué

For slang:

toxico, camé

If one wishes to express addiction to a specific substance, the -mane suffix is often used in formal language:

héroïnomane, cocaïnomane, opiomane…

Or in slang, accro à:

accro à l’héroïne/la cocaïne/l’opium