Its sounding like you don't have a good ground connection from the battery. Measure the voltage from the positive terminal of the battery to the chassis (not to the negative terminal). Even if that looks OK, you still need to check and clean both ends of the negative cable and their connection points. Best to do the same with the positive cable. After cleaning them use some petroleum jelly (Vaseline) to coat both surfaces before reconnecting.
If your cables are old, then consider replacing them. They can get corroded inside the casing and thus can't handle the amperage they need to. New cables are cheap insurance.
It could also be low water levels, so check them too.
