Questin about the negative form of verbs with たい

HI all,

I have a question about how to do the negative form of verbs in the たい form (I want to do something).

For example: I want to eat 食べたい

I learnt that the たい form is used like an adjective in い, so I usually make the present tense negative changing たい with たくないです, so the sentence "I dont' want to eat" becomes "食べたくないです".

But today I found the same sentence translated as "食べたくありません", that is using たくありません instead of たくないです . So my question is, in first place, if this translation with たくありません is correct or not, and if it is correct I'd like to know if there's a difference of meaning between the two translations or if they're just the plain form and the polite form (but in this case たい doesn't seem to behave like an い adjective anymore, I think).

Thanks!!