It is the general position of this board not to travel with old countries' passports. I have not heard of anybody actually doing this, but it could potentially work. However, you must be aware that a U.S. citizen is expected to enter the country with a U.S. passport.
Now, I think the best should be to insist with the passport agency. When you have a trip in less than two weeks they should take you and process the passport in time. There are quite a few posts in this board about this. I haven't done it myself so I can't say if the best thing is to call to make an appointment or stand in line. To be honest I wouldn't risk travelling with the old passport, even if no visa is required. I believe if you make the effort you'll get the passport in time.
PS: On reading your message more carefully I see that you already have an appointment for next week. I believe they can give you the passport next day after the appointment, perhaps the same day depending on your needs. You should be able to get the passport in time.
