My headlights get brighter when the engine is.

The battery is the key component in helping hold the charging system voltage to a safe level. It can allow the voltage to go too high if it has a loose connection, but it can not cause the voltage to go higher. Alternators are less efficient at low speeds. Brighter lights when raising engine speed proves it is working. The voltage regulator controls how hard the alternator works to meet the demands of the electrical system and to keep the battery charged.

The first thing you need to do is measure the battery voltage with the engine running to determine which state is normal, low speed or high speed. The voltage must be between 13.75 and 14.75 volts. If it stays within that range at all engine speeds, the system is working properly.