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The voltage you measured after starting is a sign that the charging system is working.
If your battery is bad, you could find that when idling, when the charging system is not doing much and the battery is needed to maintain voltage to the ignition system, the voltage drops and spark quits.
Watching the battery voltage while it's running and when it dies, as Edward-Hubert suggested above, could tell us some useful information.
Also, when you get home tonight you could put a meter on the battery and turn on the ignition switch to put the load of the tail light (etc) on the system, and see what the voltage does (engine not running).
Another battery test would be to take off the spark plug caps and ground them to the engine. Then crank for four or five seconds at a time, letting the starter cool between cranking, and see if it can do that several times without getting slower and slower.
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