I am a UK/US dual citizen. I came to the states on an ESTA. I'm aware I can only stay in the states for 90 days on the ESTA. I am getting my American passport while I'm here. Can I then use my US passport to somehow extend my stay here?
The existing answer is a bit short, so I'll answer your question line-by-line.
I am a uk/us dual citizen. I came to the states on an ESTA.
If you're a US citizen, how did you enter on ESTA? You're a citizen already.
I'm aware I can only stay in the states for 90 days on the esta.
On ESTA, yes. But if you're a citizen of the US already, ESTA is irrelevant.
I am getting my American passport whilst I'm here. Can I then use my us passport to somehow extend my stay here?
A passport is just a document showing your citizenship to others and other countries. If you're already a US citizen, you don't need a US passport inside of US borders. Any documentation verifying your identity (and showing your US nationality) (for example, a driver's license) should be enough. Can you extend your stay? Yes. You're a citizen. You can stay as long as you want.
The other answer is correct, but I'll add some relevant code citations. Legally speaking, your stay in the United States is not limited.
First, you entered with ESTA. This means that you were nominally admitted under the terms of the visa waiver program, 8 USC 1187. This provides for a waiver of the requirement of 8 USC 1182(a)(7)(B)(i)(II). That is the requirement that a nonimmigrant be in possession of a nonimmigrant visa.
But what is a nonimmigrant? That is defined at 8 USC 1101(a)(15):
(15) The term “immigrant” means every alien except an alien who is within one of the following classes of nonimmigrant aliens—
Before we proceed to look at the actual classifications, we should look at the definition of alien. That is found at 8 USC 1101(a)(3):
(3) The term “alien” means any person not a citizen or national of the United States.
But you are a citizen of the United States, so you are not an alien, and none of this applies to you. Your stay in the United States was never limited, and all you have to do to prove it is to prove that you are a US citizen. You can do that with a US passport, or, as noted by Mast, with any other evidence of citizenship.
I would add that your entering the US without a valid US passport did violate 8 USC 1185(b), but fortunately there is no penalty for that.