You have the classic symptoms of a bad regulator, but also look carefully at your battery and connections.
At least in some voltage regulating circuits the battery provides a reference voltage and base load. If so, even a good regulator will not work properly. This is why you should never disconnect your battery while the car is running. With very poor connections the battery will not provide a good reference for regulation (though often the car won't start either), and as Klaus states other parts can be damaged.
Whether due to a bad regulator or not, if the overcharging went on for a while (like a trip of several hours) your battery may have got very hot and boiled out a large fraction of its electrolyte. If a sealed or maintenance free battery, it could even have a damaged or leaking case from pressure. If it is a vented type you may need to add water. This is based on experience.
Don't mean to preach, but when checking the battery please shield your eyes - especially an overcharged battery may emit lots of hydrogen gas. See:
http://www.sightandhearing.org/sightcenter/battery.asp
I'm personally aware of 2 battery explosions though neither happened to me.
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