LB's right, sounds like an ignition problem. A new switch is cheap ($15 US) to see if we're right. Why the new battery? Weak or dead? Could that have caused some of the glitches your experiencing? If your battery was weak you think the fresh battery exposed a weakness in the switch?
I know, more questions than answers.
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Sadly on the outside looking in.
"Drive it like the Doctor ordered"
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