My single phase power panel works fine, as long as all of the 240v breakers are off. Service into the main breaker reads a steady 120v on each leg from the transformer, and 240v across both legs. But, when I flip on ANY of the 240 breakers, all 240v jumps to one leg of the panel (for both single pole and double pole breakers), and the other leg goes dead. This is causing many problems throughout the entire system... 240v to one side, and 0v to the other for all circuits. As you can imagine, we are having multiple overvoltage/dead circuit problems. As long as the 240v breakers are off, the system works fine.

After disconnecting everything in the panel, the continuity readings all come back clean, indicating that there are no crossed or broken wires.

Any suggestions or answers??

Sounds like a neutral problem, Measure between each hotleg and neutral(should be 120 volts) and between the 2 hotlegs(should be 240 volts).
Start at the Disconnect, then lugs on panel, then out of breakers last.
Could be a Power company problem.
Neighbors having problems?
Take a visual at your overhead wires and out to pole, an look at the bare wire for a broken connection. You can sometimes see the problem.

Come back clean? You mean open, not shorted, right? If it's a system wide problem you could have an open neutral... that would give you the same reading. You said ANY of the breakers? But all of your 120v outlets work on both sides of the panel.. unless you turn on a dbl pole... is that correct? I'm wondering if the neutral / ground bond isn't completing the circuit when the dbls are off. Might be time to call an electrician. Do as Strat suggested first, check with the neighbors and do the troubleshooting as long as you have the experience to do it.