Jetta 3 1.8 1993 model

check the fuel pressure regulator for correct operation

Have u put it in Park and press and hold accelerator to see if it still cuts off. How long of a drive before it cuts off

most problems like this are caused by these things 1: Crank sensor 2: Cam sensor or if you have a distributor pick up coil. One of them items is most likely you problem. You could also check your coil but that is less likely to be the problem.

Yes, but remove the battery, and take it to any parts store, as most will test it for free.

plug wires for sure are bad with it backfiring

mcdevito75 here, It"s possible the alrm system and the back fireing are associated, specially if the alarm (security system) is factory (built with the car) BEST BET have an auto alarm shop take a look, most auto shops that do glass, windshield, doors etc. also handle alarms.

well if u have the 4.9 it may possibly be the valves seals leaking after they warm up they swell once you start drivin it burns off. the flow of the presure keep from seapin. now if it is white smoke you got water leakking in somewhere and you need to seek immediate help

run the fault codes
check for problem wiring or crank/camshaft sensors

will it start back up

I heard of an issue with the fuel pick up in the tank. It has a check valve that gets hung up or freezes in the colder climates.