I know Egyptian pharaohs has cartouches but did they have signatures? Seeing King Charles having to sign things, did the kings of Egypt have to do that? When the amarna letters were sent did the kings use a signature?

Not as far as I’m aware. They didn’t write their own documents, and Egyptian documents (I can’t speak for the letters written in cuneiform) are extremely formalised when using official language. Cartouches weren’t exactly used in Hieratic (there’s a couple of signs that go either side of it but not fully encircle) because their role is to make the name endure for eternity. Looks like this in transcription and Hieratic (these are the same name, one in hieroglyphs which is the transcription most Egyptologists translate from, and the other the original Hieratic written by a scribe):

[id in alt]

So, no, there’s no such thing as a signature for an Egyptian king. I’m actually not sure if we have anything personally written by an Ancient Egyptian King.