Anytime these plans come up, they're light on details. But I think what they're proposing is using the LIRR ROWs, but building new dedicated high speed track on top. So adding new high speed track at ground level where possible, and maybe a viaduct above LIRR tracks in other places. Or something like that. Again, they're light on details, so I'm guessing.
That would solve your bobbing and weaving problem for sure. But this might be even more fantastically expensive than it looked at first glance.