No US citizen should ever get ESTA. That is for people who do not have any US citizenship.
Citizens should enter their own country with their own country’s passport. Therefore you should enter the US with a US passport and enter the EU with an EU country’s passport.
Airlines are concerned with the possibility of you arriving at a country and then being refused entry. They pay fines and penalties if that happens, because airlines are expected to pre-check your paperwork. When they see you are a citizen of the destination country, that puts them at ease because you cannot be refused entry if you are a citizen.
America doesn’t have exit controls so they don’t care how you leave. When boarding a flight to the EU, show your EU passport.
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.
Credit:stackoverflow.com‘