A faulty or intermittent fuel pump can cause the motor to quit. It has nothing to do with the tell tale water flow.
It is possible you have more than one issue with the motor. The tell-tale stoppage should be looked at right away as described in an earlier response.
A dealer with a factory trained technician would be your best source to troubleshoot the motor.
"There is never just one thing wrong with a boat";
-- Travis McGee, main character in a book series by John D. McDonald
The factory recommends that a properly trained technician service your Johnson or Evinrude outboard motor. Should you elect to perform repairs yourself, use caution, common sense, and observe safety procedures in the vicinity of flammable liquids, around moving parts, near high-temperature components, and working with electrical or ignition systems.
The information offered here is only general in nature and should not be construed as complete factory approved procedures, techniques, or specifications. Always use the proper service manual for your motor, up-to-date service literature, the correct tools, and have an understanding of how to proceed with troubleshooting and repair methods. If you are unsure or uncomfortable with a procedure, a situation, or a technique, enlist the services of a factory trained technician.