Your link to the RCGroups post seems to be trying to put new firmware onto the ESC's. Did you ever do that successfully so that your ESC's now have new firmware on them? Of course how ever you do that, it is permanent. Once the ESCs are fixed, you can fly any firmware (CleanFlight or LibrePilot) on the flight controller. ESCs should be recalibrated/reneutraled after flashing.
If the problem happens even with props off, you can take props off and test in LP output page. Click the "Link" box for each motor. Click the "Live Testing" box at the bottom. Drag the sliders left and right to test the motors. If they work well here, but don't work well in flight, you might have a vibration problem. Do NOT click Save after doing this testing or you will have to reneutral your motors again. If you must leave props on, then please be safe and hold the model firmly and move the slider carefully. In that case it is probably best to not click the "Link" box, so that slider only makes one motor move.
If increasing Stabilization -> Expert -> GyroNoiseFiltering helps then you have a vibration problem.
Also, you might try PWMSync (ESC protocol) or PWM490 because I recall there at least were some OneShot incompatibilities in the past.