Continuity test across a battery? Doesn't sound like a useful test to me. If battery is dead, and it shows a short, well it would. Batteries are supposed to e very low resistance, that's why the voltage doesn't drop when you apply a load. Test cross a charged battery? Smoke the meter.
Now, continuity, or more useful, resistance from end to end on cables, that's useful. Across contacts on the switch too. Leave the meter clipped to the cables and watch to see if the reading changes as u shake the cable. Feel for any hard spots that don't flex as easily... That's corrosion increasing the resistance...
Yes, I know u said the new batts seem to have solved it, but it's still possible u have only fixed the symptm, not the cause... And she will let u know at the worst possible moment...
Had a car once thatwouldnt turn over at all once in a while... Until I kicked the tire.. finally found the main chassis ground wire corroded and intermittent under the engine somewhere.. not the only car of mine that had an intermittent wire make the thing act like it was possessed