My car dies after its bein driven for a lil while

my chevy 1998 chevy corsica starts right up once driven for a lil while and shut off it will start up and then start sputtering and eventually die. However it seems to keep running if its not shut off.

How old is the fuel filter?

Is the engine reaching correct operating temperature? Does it warm up quickly or slowly?

just replaced the fuel filter and there hasnt been a noticeable difference in the heating and cooling. thinkkn ite b da fuel pump