Can I say, "Necesito callarme?" I need to shut up in Spanish? Does it make sense?
I say this when I talk too much! Are there any expressions I can use?

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Yes, of course it is correct, and also you can say "no debo hablar más de la cuenta"

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DEBERÍA CALLARME, ESTOY HABLANDO DE MÁS.

This is what I would use in such case

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According to Spanish grammar, this sentence would be perfectly constructed. However, you're saying something like " Do I need to shut up?" , and that's a little bit harsh. I think that you're rather aiming to say something like " Should I speak less?" . That sounds great of Spanish, and you can perfectly use  " ¿Creen que debería parar de hablar tanto?"/ "Creen que debería dejar de hablar todo el rato?"  
That sounds very good, and you can avoid lots of misunderstanding, just because you can use those sentences both in formal and informal situations.
Cheers!

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Yes, you can say that. "Suelo hablar demasiado, necesito callarme".