If either the battery is not grounded to BOTH the engine and the car body; or the battery is only grounded to the engine, and there is no ground wire from the engine to a bolt on the firewall; this can cause such problems.
The battery should have at least 12.4 volts across the terminals when the engine is not running. If it is less than 12.2 volts; the battery is either defective, or it is not being properly charged by the alternator.
Another possibility is that the alternator is not producing full power. This can sometimes happen if a diode in the alternator goes bad. This can be a tricky thing to test; because if you check the alternator voltage with no accessories running, it will often test good. The sure way to check the alternator output is to have the engine idling; and turn the headlights on to high beam, and turn the heater fan on high speed. Then check the voltage at the battery, with a digital voltmeter; it should be NO LESS THAN 13.7 volts under that condition. If it is lower than that figure, the alternator should be replaced.
If it is not a missing ground cable, and the battery and alternator both test good; then the ignition switch has developed excessive internal resistance, and is now just barely able to carry the current needed for the ignition system (which is why the engine falters when additional electrical accessories are turned on). In that instance; the ignition switch's electrical section must be replaced.