Hydrocarbons do tend to oxidize. However the heavier ones (more atoms per molecule) don't so much [edit:readily in room temperature environments].
"Flammable", besides the lay meaning of being easily set on fire, is actually a
techincal term. While hydrocarbon fuels are typically flammable, they can also be
combustible. Diesel fuel sits around the borderline or above, depending on the particular mixture. For cold weather they mix in more lighter molecules to keep it from turning to jelly in the fuel tank.
Also jet fuel is usually NOT flammable. The reason is that by general rule the heavier the hydrocarbon molecule the more energy/weight it has** and the more controlled the oxidation (in gasoline this is refered to as having a higher octane). Because so much of an airplane's weight is fuel and weight is so important to an airplane, the trick to jet fuel is to have heavy of molecules that still burn completely so you get all the energy out.
Of course there is also the issue of safety in the case of a crash it's better to be riding with tank of fuel that is much more stable.
** This is really about the ratio of hydrogen to carbon bonds vs. carbon to carbon bonds, the later having more energy, and the total number of bonds in the molecule. But those details are for physical/organic chemistry class.