When I turn off the car, the light for the clock/radio stays on. I just changed my battery two days ago and would hate for it to drain because of it. What do I do? I doubled-checked that all the doors were closed properly and re-started the car ... nothing worked.

The LED light would only use a tiny amount of power to stay lit, much less than any small bulb in the car. You might have a poor battery, or you might have a drain on the battery from something else staying on. Any parts store could test your battery for free. That's where I would start, making sure the battery can hold a full charge.
Is the clock light in a radio display? Maybe an amplifier or something connected with the radio is staying on? If the clock is a separate fixture (and I don't even know if they still put such things on cars lately- a separate clock, I mean), then the clock could be unplugged. I don't think there would be a way to turn the light off.

If you have a known good battery and it is still draining fast, there are YouTube videos and explanations on how to hunt down an unintended draw. Mechanics may refer to it as a "parasitic drain"
but it is really an unintended drain. Your car's computers, and security systems, and radio and clock memory will always require a very small amount of battery power when the car is shut off, but that amount is measured in milli-amps, very small fractions of 1 amp of power. usually less than 100 milliamps which is 0.100 amps, or 0.1 amp- see, much less than one single amp. A small bulb like under the dash may be rated at around 2 amps or up to 5 amps for typically larger bulbs. A small bulb might actually drain the battery over several hours. Headlights , you are talking lots of amps, I'm not certain but say around 20 amps. So you see, your clock light would have to have a bad short in the wiring to actually drain a good battery. The light itself would not. This is just me talking off the top of my head, I'm not an expert on electronics or electrical circuits, but I think the information is accurate.

Sorry to say, but none of these are related. Fuel gauge is probably the sending unit in the tank, could be wiring. The air bag could be the module,the bag unit, the sensors, or bad wiring. The radio is probably your most likely choice for battery drain. The three main things I have found to drain batteries are the radio, the starter, and the alternator. These three will kill a new battery in 1 to 2 days, An older battery over night. Shorts in the interior lighting is 4th..

sounds like the radio it's the problem. to make sure it is, you have to do what mechanics call a parasite circuit test, simply buy a test light at your local auto parts store, open the engine hood and look for the battery disconnect the negative cable and connect the test light in series with the battery and negative cable, the test light will stay on indicating that in fact there is a parasite circuit drain, find the fuse box your car comes with a owner's manual this manual it's helpful to find the fuses in the car, now that you find the fuses take out one fuse at the time until the test light goes off, (important put the fuse back before moving to the next one) when the test light goes off by removing the fuse, that is the circuit that it's draining the battery, check the fuse number whit your owner's manual to see if the radio in fact it's part of the fuse circuit. if not then you have a drain somewhere else in that circuit, (important the car doors must be close while doing this test or the interior light will stay on causing a normal drain keeping the test light on. and if the test light does not come on when connecting it in series don't panic this means the drain it's too small and then you have to do the same test but this time using an amp meter which it's more expensive and more complicate than the test light.

Change your blower motor resistor,it's located next to the blower motor.

There is an electrical short in the system. Have it checked by a professional.

Check the connecter of you battery. There's a good chance your battery connecters are loose, or another problem with your battery connecters.

make sure your DOME light dimmer switch is not on.If light still on the radio, then remove the 10 amp fuse for the radio.If light goes out you can either replace the radio or re-install the fuse each time to listen to your radio. there is a short somewhere causing the radio light to stay on.

same thing is currently happening to me.. please tell me theres a solution.

your radio is staying on? you have a bad door switch.the bcm does not see door opening and thus does not turn radio off. most likely the drivers door.

pull the courtesy fuse temporarily and see if the battery still drains,if it does then it may be in the alternator