Turn signals stop flashing when I step on the brakes

I'm not familiar with your car model, but what you described is the classic symptom of a broken ground circuit. What is not common is that both sides are affected at the same time. It is possible both rear light assemblies share a common ground on your car.

Where I would start is definitely not with cleaning anything. The huge goal here is to NOT solve the problem until we know exactly what is causing it. If you clean contacts, wiggle wires, even unplug light bulbs and slide them back in, those things can mask the cause of the problem. Then, when the circuits are working, you won't know if the problem is solved correctly or permanently.

Use a grounded test light to measure the voltages on the socket of one of the malfunctioning lights. Most rear lights have two filaments so they'll have three terminals; one for each filament and their common ground terminal. With the newer flat plastic-base bulbs, you can usually poke the test light's probe into the socket and reach the terminals. With the round brass-base bulbs you may need to slide the probe into the back of the socket next to each wire.

Turn on the running / tail lights and one of the turn signals, then take the three voltage readings. You should find 12 volts all the time on one terminal, 12 volts pulsing on and off on another terminal, and 0 volts on the third one. If you have a bad ground, you'll find 12 volts on the two terminals, and something higher than 0 volts on the last one. Typically that will be 2- 6 volts, so the test light may be very dim. If you do find a terminal where the test light doesn't light up, it would be smart to recheck it with a digital voltmeter. That will show voltages too small to make the test light light up.