I 1000% agree. Email is an impersonal, unreliable means of communication. For a matter this important, you need to speak with someone. Texting or calling the right department should get you an answer.
I will say that Princess’s response to dietary requests has changed since I first sailed them a year ago, or so it seems. In March 2023, I had a postcard on my stateroom bed on day 1 to bring to the MDR to initiate the food sensitivity/allergy conversation with the headwaiter. On my next cruise in September 2023, I had nothing but my previous experiences to guide me.
My issues are not life threatening. If they were, I might share the OP’s anxiety. But I wouldn’t give up after no response to an email. I’d try a different approach. You know what they say about doing the same thing over and over and expecting a different result.