My car did the same thing. With the lights off and running it was about 14 steady. With the headlights on it varied wildly between 13.8 to 14.2, which explained the flashing lights.
There is an unused bolt hole located in the back of the passenger side head. I attached a heavy duty copper strap with a tin coating from the head to the blower motor stud. The voltage is now steady with no flashing lights. Extra grounds increases the reliability of any circuit. My car must have been hunting for a ground under load.
The other possibility is that the battery has blown out one of the diodes in the alternator. The alternator will charge, but when placed under a load it begins to give up. A Battery can fail in a number of ways. Intermittent shorts between the cells is the worst. A battery in this condition may start the car, but as the battery vibrates the sulfates can partially bridge the positive and negative cell grids and hammer the diodes in the alternator. If I remember there are 3 diodes in our alternators. By the way, the diodes are now made offshore as far a I can tell.
My suggestion is to remove the alternator and have it bench tested at a shop. If it is good then try the ground strap.
The other possibility is a loose wire or connection. Using a volt meter with a needle, otherwise known as a D'Arsonval movement, attach it to the battery and start the car. First observe the reading with the lights off. Then turn on the headlights and high beam and very slowly and carefully disturb the connections at the voltage regulator, alternator, and ignition key plug under the dash to start. Your assistant must watch for any change in the meter movement. If the assistant notices a change they yell STOP! At that point you have found a problem. It may not be the problem you are looking for, but it most likely will be. If your assistant is not diligent in observing the meter, don't bother doing this test. Be very gentle on the wiring, just a little tug and wiggle, ever so slight, is all you need.
Mechanical voltage regulators can exhibit this condition. I wouldn't worry about it if that's the case. If you have a solid state regulator then try the aforementioned tests. Good luck.