You have this correct:
We could watch an interesting series or anime.
This is the usual way to refer to a choice from the group of interesting things which are series or animes.
But it will be ambiguous if plural (so the article is absent) and will be understood on the sense of the particular adjectives and nouns:
Look at the big buses and taxis
Look at the noisy buses and taxis
Here the buses are big and the taxis are not; while the buses and taxis are both noisy.
We could watch an interesting series or an anime.
Formally this means the series is interesting and we are saying nothing about the anime. In practice this is ambiguous and depends mostly on the sense of the particular adjective and nouns and any emphasis when spoken.
A native speaker might well respond "Are you saying animes aren't interesting?" To which you might say:
We could watch an interesting series or an interesting anime.
This is a correct sentence which you would use only to specifically emphasise that both are interesting.
If you want to unambiguously apply the adjective to only one, put it on the second noun:
We could watch an anime or interesting series.
This doesn't necessarily mean the speaker considers animes to be uninteresting, just stressing that the series for consideration must be.
... or use a balanced construction which alternates both the adjectives and the nouns:
We could watch a funny cartoon or romantic comedy
Here it's clear the cartoon is specifically funny while the comedy is specifically romantic (though funny too, it's a comedy).