Your description, including the tower thing (although I don't think you need to land on it) and the aircraft carrier, reminded me of
Evasive Action, which indeed has a playable demo.
If it's not it, you can take a look at what other DOS combat flight simulators that had playable demo or shareware versions I managed to find:
http://www.classicdosgames.com/forum...php?f=2&t=1125