As a US Citizen - which is what gives you the right to a US Passport, you can stay indefinitely in the US. You are not even required to have a US passport, although that makes it easier to enter the United States. Your passport is valid the day you arrive, so there should not be any problem. You could be coming back to renew your passport as I suggest you do, to make it easier to get back in. Should you not have enough time to do so, you can always have it renewed at a US embassy abroad.
Given that you are a US citizen, you are also not eligible for an ESTA. Leaving on any passport is not an issue in the US because the US does not do exit immigration. Airlines do share data telling who is leaving the US but that is not the same as exit immigration.
Given your schedule seems a little tight, just procede normally. Enter the US on your still-valid US passport then fly out and enter the EU with your EU passport. From there you will have more time to renew your US passport which you can do without the emergency procedure. Just make sure you have all the documentation with you when you leave the US. The US government has all the relevant information on their website here.