Before addressing your actual question, I need to address something else.
To me, one problem with your sentence is putting the adjective directly after the noun without a linking is verb. This makes it sound almost as if different could be considered in an adverbial sense.
For instance, Apple deliberately broke accepted rules of grammar with their now famous slogan of think different. Prior to this, if you'd wanted to use an adverb, you would have said think differently. But since it's worked itself into the unconscious awareness of some people, it's now less obviously wrong when it's used in an adverbial sense. (Although, personally, I would still change it.)
So now, thanks in part to Apple, when I read your sentence, I find it partially open to that interpretation:
❔ I drink a coffee different from other people.
This could be taken to mean one of two things. I will categorize the different functions of different and provide alternate phrasing:
As an adjective:
I drink coffee that is different from other people.
I drink different coffee from other people.
As an adverb:
I drink coffee in a manner that is different from other people.
I drink differently from other people.
In your question, I am assuming you mean different to be taken in an adjectival sense.
You don't like either of these sentences:
I drink a coffee different from other people.
My coffee is different from other people's.
I suggest clarifying your meaning by moving the noun to the front of the sentence, switching to the definite article, and inserting that of before people:
The coffee I drink is different from that of other people.
Using that of ensures that you're talking about an item in relation to other people, not about the other people themselves. And, in context, it's clear that the thing related to other people is the coffee they drink.
This construction also works with paintings and politicians:
The painting I like is different from that of other people.
The politician I like is different from that of other people.
Note that this does sound slightly awkward. Normally, we would not use this kind of phrasing—instead, we would repeat the verb, the noun, or both. But you've specifically said that you do not want to repeat either. Given that neither can be repeated, this seems like the least ambiguous way of writing the sentence—and a method that can apply to any subject.