Normalmente, bebo una margarita pero tienen una bebida nueva se llamada “naranjada de pastel”.

We say "tienen" because we're referring to the bar as an entity where multiple people work, so it's "they have" and not "it has", just like you wrote in the native text version. If you want to refer to the *menu* having a new item, you may as well use the impersonal verb conjugation "hay" (the infinitive is "haber") → "hay una bebida nueva".

Regarding the second correction, if you'd like to use the exact same structure as in English, you'd keep the correction I made → "llamada" ("called"). If you want to use the verb "llamarse" instead, put "que" ("that") before "se llama".

You can also omit the comma after "normalmente", in oral speech we wouldn't make a pause there.

He teniprobado esta bebida antées y está muy buena.

"To have a drink" doesn't sound very natural when literally translated into Spanish. You can say "he probado" ("I have tried") instead, or just "he bebido" ("I have drunk").

Moreover, when we say that a drink/food tastes good, we use the verb "estar" instead of the "ser" you used.