There can be up to a 10 minute delay between a call for cooling and the outside unit kicking on. Also some rheem units have a pressure switch on the outside unit located near the bottom that is usually red. try pushing this switch in. Does the fan come on in the furnace when the thermostat is set to fan on?
Also, what lights are you referring to when you say they go out? LED's on the furnace control board?
are both outside units on the same circuit?
pull the disconnect to one of the outside units and try to set the thermostat to auto. If it trips the breaker try the other one
one of those units has a short and is tripping your breakers
120? you should have 230v outside at the units
maybe something is going on in the main breaker panel to the home?
try turning all the breakers off and then on again
are you checking voltage at the disconnect? between both legs you should have 230volts
if you have 120 at both units it tells me that the main breaker in the home may have something wrong with it
turn the main breaker to the home off and back on. turn both furnaces off at the furnaces, and then recheck the outside voltage please
do you have 230 volt dryer outlet you can check your meter with?
what voltages did you read at the dryer outlet?
is the stove gas?
If you have two separate outdoor units that are both only reading 120 volts that tells me there is an issue inside your breaker panel. the main breaker feeds 2 legs of 120 volts to give 230 volts AC. If your meter is accurate either the main power coming into your home has lost 1 leg of 120 volts or the main breaker has an issue
At this point I would recommend getting an HVAC tech out to check your main breaker panel
your welcome. glad i could assist