How many miles? Big difference between 100k and 200k (regarding where to look first). If its high miles 200K+, I would replace the CPS crank position sensor. It needs one anyway if its high milage. Mine recently went and gave me a rainbow of symptons. Everything from hard cold starts to shut downs after warmed up.

The CPS gets weak and tired as they get older. Im not sure but I believe weakening magnets may effecting the sensitivity of the pickup. Oil leaking on them from the filter/sending unit also wears them thin and may effect resistances. I believe the replacements are improved (mine was a little different). Well worth the $140.

The fuel pressure regulator is vacuum controled and either increases or decreases flow when told to do so. Perhaps it is stuck at increased flow(?) position and needs to be replaced (flooding). If the accumulator is bad, I think it would starve for fuel.

You may want to put a cap and wires on it also if they look tired.

More and more I stand by the rule that whenever a high mileage 124 is purchased, the buyer should replace the below original parts with new...

OVP
MAS (fuel pump relay)
Both pumps
Accumulator
Fuel filter
Primary fuel line (2' long). They get rusty at the bottoim curve.
Tank Vent
Crank position sensor
Oxygen sensor
Fuel pressure regulator
Oil presure sending unit
Plugs, wires, cap
Coil

If you do the work yourself it will all cost around $900 low, $1200 high,. When you are done, you can drive that car coast to coast for another 200K.