I am having issues with my car

What exactly is cracked on the battery? If it's the plastic case, of course it must be replaced to prevent acid from leaking out and destroying paint and wiring harnesses. If a post is loose, that's from someone pounding on it or twisting it. Acid can leak through that gap.

The battery light is telling you the battery is not being recharged while you're driving. That is usually the fault of the generator but wiring problems can also be the cause. Have a professional load-test performed on the charging system. Many battery stores will do that for you for free. On some car models that light will also warn of an over-charge condition. That can easily happen when there's a loose battery cable because the battery is the main component in helping the voltage regulator maintain a safe voltage.

You can do a preliminary test with an inexpensive digital voltmeter. Measure the battery voltage with the engine off and with it running. Off, it should be 12.6 volts if it's fully-charged. If it's around 12.2 volts, it's good but discharged. If you find it's closer to 11 volts, it has a shorted cell and must be replaced. With the engine running, the voltage must be 13.75 to 14.75 volts.