Turning key will not even turn engine over. Use jumper starter to start. Put park brake on to keep running. As soon as I disconnect black jumper cable engine shuts off. Occasionally engine will continue to run, but then when I try to engage the blade, engine shuts off. If I just touch the battery post with the jumper cable, the engine keeps running. Could this be simply the battery being bad?

It could definitely be a bad battery. Or just a severely discharged battery. The charging system does not charge the battery very fast. You could try to put the battery on a slow charge and see if it comes back to life. Otherwise, a new battery is only around $30-35.00