Since your two sentences mean almost exactly the same thing, I don't see why you need to distinguish them. But a slight variation might be more illuminating:
I got my car so I can give you a ride.
(I walked a mile-and-a-half home and drove my car here in order to give you a ride.)
I (have) got my car, so I can give you a ride.
(I just happen to have my car here, and I'll be happy to give you a ride.)
The answer is that the only way to distinguish these two sentences in informal American English is by context (or in this case, by hearing the very slight pause indicated by the comma).
I can see two ways to parse the phrase "have him get addicted to drugs", in different contexts.
The first means "Cause it to pass that he gets addicted to drugs".
This matches exactly with "Have him get killed" in the other question, which as I said would be unusual in reference to reality rather than in terms of a writer's relationship to fiction.
Still, it would not be impossible in referring solely to real people, it's just unusual because in circumstances where we could arrange such a thing to happen we would have strong control over they means and it would be more likely to express that as "have him killed".
Now, here we could say "have him addicted to drugs", but the adjective form of addicted is much more common than the past participle addicted so it reads strange for that reason.
We could say "have someone addict him to drugs", but again that sense of addict isn't commonly used. Let's assume that most people wouldn't even think of that sense.
So therefore, "have him get addicted to drugs" is more normal. It's the same usage as "have him get killed" except that there are more normal ways of rephrasing the latter than the former, so the latter is more likely in the special case (relating to creating a narrative) only while there are no such alternative phrasings for the drug case.
There is another possible parsing:
Live with someone for a few years, raise beautiful children with them, then have them lose their job, then have them get addicted to drugs, then have them suddenly become violent like they never were before, and only then would you be able to talk about what you would do if you were me.
This phrase from a (thankfully fictional) speaker who has been through the experience of living with someone who is addicted is putting forward a hypothetical. Here it uses the verb have in its sense of "experience, undergo", in an imperative mood that doesn't make literal sense (they are not suggesting someone actually do this) but could relate to something that does (they are relating it to previous experience).
I'm sure there are other ways to parse it too, as you only give a bare phrase rather than the amount of context you did with the last question.
Best Answer
First of all, It's usually "I've got it". But that's just nit-picking.
Native English speakers usually use either interchangeably to mean the same thing, that is, they understand now. There doesn't seem to be a difference in meaning or usage due to the different verb tense.
They also sometimes add "now": "I've got it now" or "I get it now".