Measure the main voltage with the engine running at 2000 rpm or so. You should see the system charging to somewhere between about 13.9 and 14.4 volts (higher value at lower outdoor temperature). Measure this at the battery terminals with an accurate meter. Put on high beams, heater fan to high, etc to provide a significant load on the system. If the test fails (voltage too low or fluctuates) look for loose connections at alternator/regulator. One possibility as the cars age is dirty slip rings or worn brushes in the alternator. Also possible the charging system is working but you have a partial short circuit or other overload.

The low coolant light does look a little like a battery light - I've had a person unfamiliar with my car make such a mistake. If the voltage is fluctuating you are very likely to hear the fan whine as you report, or see the lights dim time to time.