Engine RPM erratic and shutting off while driving

So maybe it could be the fuel pump again because the fuel pump I got my dad got it from the junk yard

I’m having the same Problem what is this that broken off

I have a 2003 Impala. It started to "stutter" when it was shifting gears automatically a few months ago but only very sporadically. If you stopped and turned the car off for 10 seconds it would start back up with no problems thereafter. A few days ago it would start right up but when put in gear dies immediately. After a good many tries, it made it to the neighbors house roughly about 1/4 mile away without anymore problems. On my return trip it died half way back and wouldn't stay "on" no matter what I did. The next day my husband went to check it out and it started & drove without any problems. He changed the fuel filter and checked gauges, etc but it seemed to work fine for him. However the next day it left me on the side of the road again after going 1/2 mile away from home! It seems to only go 1/2 mile at best before it warms up and then dies and presents the same problem again. It starts fine but doesn't want to idle at all. The only way to get it going now is to put in neutral, start car with gas & brake pressed at the same time. Is my car possessed? What could be wrong?
Thanks!