There are two kinds of failsafe, RC (radio control) and FC (flight controller).
RC failsafe uses features of the radio and the FC never knows the radio is out of range since it continues to get servo pulses from the radio. You set the radio to do exactly this and this with all the channels. For your desire to have it drop out of the air (which is the best unless you have GPS ... correctly configured GPS flight modes and failsafe can make it fly back to where it took off) typically the only thing you must have is for the throttle channel to go to zero. You must read your radio manual to determine how to set up and use (RC) failsafe, and what kinds of (RC) failsafe are available in it.
Part of RC failsafe, and usually the radio default setup is to "hold the last known channel positions". This is the worst possible setup as it guarantees a fly away as soon as you get out of range. Some radios only have this and there is no way to change it. That is the worst. From what I think you said, this sounds like what is going on so you should look for a way to enable the normal RC failsafe, or for a way to disable servo pulses when out of range so you can use FC failsafe (see below). I suggest Google...
FC failsafe is where the radio is configured to stop sending servo pulses when it flies out of range. In this case you set up the FC to know the pre-programmed failsafe channel positions and the FC switches to them when it sees the servo pulses go away.
Of course one thing to remember is that the FMS (flight mode switch) is one of the channels. If you have a Revo class FC with a GPS, the obvious choice is to configure either kind of failsafe to put the sticks in the middle (particularly the throttle since zero throttle can mean "motors off") and set the FMS to RTB (return to base = (by default) return to takeoff location = called return to home in some places).
One time I set up both RC and FC failsafes to figure out what was causing unexpected failsafes on a quad. RTB configured in both cases, but RC failsafe programmed with a little left yaw and FC failsafe programmed with a little right yaw so during RTB it would spin one way for RC and the other way for FC. Note that RTB and waypoint quad GPS flight modes allow you to do what you want to with yaw and the quad just flies backwards or sideways as necessary to maintain a straight line back to the takeoff location.