Is it my alternator or battery?

I have a 99 ford crown Vic police interceptor. This happened to me twice. One day I was driving and I was pulling into the auto zone to go buy oil. The second I turned into the parking lot my car cut off and the battery light came up on the dash board and it was really HARD to turn my car! Not even 2 week later my car did the same thing in McDonald's parking lot! Like it starts up with no problem so I don't know how it could be dead but it shuts off and making it hard to turn. I'm scared to drive it far places and not being able to turn properly and hitting someone. Help please. I read that it could be my Alternator or dead battery but my car starts fine.

It sounds more like a bad connection somewhere in the ignition or fuel systems. If the alternator failed, the engine would still run and the battery light would come on. If the battery was dead, the car would not start to begin with, but once started it would run using the alternator.

Big Fords are famous for corroding the INSIDE of the positive ( red wire ) battery cable clamp. So pull that clamp off the battery post & then clean both the inside of the clamp & the sides of the battery post with a small piece of sand paper. If that works, most all car parts stores carry a small wire brush designed just for that task.

http://ricksfreeautorepairadvice.com/battery-alternator/

Another possibility, presuming that you were not accelerating when going into the drive for the parts store, is that the engine idle is too low

Not likely a bad alt or battery. Could be a loose wiring connection or loose battery cable or other issues with low idle/IAC problem… Or a faulty inertia safety switch which some police package cars had issues with.

Try cleaning the battery cables/ends and make sure they are tight… Wiggle test the wiring from the ignition switch or other wiring harnesses under the hood for ignition or fuel injection, main relay, etc. Run the car at idle and wiggle various wiring or connections to see if the car stalls, if so, then check that wiring or connection for issues… Loose, corroded, frayed wires rubbing on metal, etc.

Stalling can also be caused by too low idle speeds especially when turning since the power steering pump adds extra engine load when making turns which the computer may not be compensating for or it does try to compensate but the IAC is not responding quick enough. Or the PS pump is defective or low on fluid which raises pump pressures/engine load above normal… Check the PS fluid level. With the car in Park and engine idling, try turning the steering wheel and see if idle drops too low or stalls… May need to check the PS steering system for issues or clean or test/replace the IAC which is a device on the throttle body that the computer uses to control idle speeds. Turning AC on should raise idle speeds slightly so that may help temporarily if it is a faulty IAC issue/due to low idle.

Another issue with police package cars was that in some braking circumstances, the fuel shutoff inertia safety switch can be tripped which will momentarily shut down the engine/fuel system although that usually needs to be reset after fully tripping. Might need to check for issues with that.

Related info…
"Date Reported
NOVEMBER 01, 1998
NHTSA Reference
#613284
TSB Reference
#985149
Description: SOME POLICE VEHICLES MAY STALL DURING BRAKING OVER RUMBLE STRIPS. THESE VIBRATIONS MAY OPEN THE INERTIA FUEL SWITCH. ENGINEERING IS INVESTIGATING CONCERN. *TT"
http://www.fordproblems.com/tsbs/Crown_Victoria/1999/

Lose belt.