Yes. You'll need to show your new passport to the airline. They will not care that you had already told them about a different passport. You may be able to update the passport associated with your booking online, before you check in.
I think the assumption that it is costly to change passport details on your record is false. When you check in, just say, "I have a new passport, here it is." In fact, you can just say "here's my passport"; Nobody cares that it is new. I do this all the time, checking in with a passport other than the one the airline already knows about; nobody even blinks.
Some "nickel-and-dime" (that is, budget) airlines charge, apparently, if the clerk has to enter passport details during check in. Or, more precisely, if the clerk has to swipe your passport through the scanner. To avoid this charge, you should visit the company's web site after you get your new passport, and look up your reservation. You ought to be able to update your passport information online, avoiding the charge.
Anyway, if you are using the new nationality to travel somewhere without a visa, you will have to show that passport to the airline or they won't let you on the plane. You need to prove to them that you have the documents you need to enter the country, because if you don't have those documents, the airline will be fined heavily for bringing you there. Therefore, if you do not show the new passport, you'll need to show a visa, which you do not have.
There are plenty of posts here and elsewhere on the internet explaining how this works. The basic rules are this:
When you check in, show the documents that you will use to enter your destination country. In the case of a transit, you might need to show more than one passport, if, for example, one requires a visa at the transit airport and the other requires a visa at the destination.
When you enter any country, show the documents that give you the best advantage.
When you leave any country, if there is passport control on exit, show the documents that you used to enter. One exception: if you are a national of the country you are leaving, you should usually show those documents, even if you entered with different documents.
There may be some exceptions to these rules, especially if one of your nationalities is of a country with restrictions against dual nationality. I have traveled around the EU and between the US and Canada filtering these principles, without any trouble whatsoever.