either or

'i don't like either tea or coffee' is a better option

I do not like tea and coffee. Compound noun; this sentence is fine.
I do not like tea and nor do ( I like ) coffee.
I dislike both tea and coffee.

I have not yet met a rule that forbids using both this way:
I don't like both tea and coffee, which means it is not true that I like both of them.
I do not like either coffee or tea means:
I like neither

I don't like both tea and coffeewould be a very rare utterance. The listener will think that you like one or the other of them, but not both, and will feel he has suddenly become part of a guessing game. Goodness only knows how he will react to that!