I'm level 18 now and the biggest plane I have available to me in the market is the Aeroeagle. It has been that way for several levels now and I'm wondering when the next model will be available in the market? Is there something more I must do (e.g., unlock a certain number of achievements) to unlock more plane types?
No, there's nothing additional that you need to do. The Fogbuster class planes will become available to you at level 20.
See the Pocket Planes wiki page on Planes for more details.