Double kitchen sinks, if one is empty the other drains fine. If both are full of water and I pull the drain plugs niether wants to drain?

Ithink this may be a venting problem and not a blockage. We have this same problem with vessel sinks in the bathroom. A regularbathroom sink vents through the overflow. Vessel sinks don't have one.
Sounds like your kitchen sink has the same problem. If one tub's empty the drain will serve as a vent for the other. If both tubs are full you've blocked off the vent and this will slow down the process. Does the trap "gurgle" when the sink drains? If so you have a clogged roof vent. Good luck, Tom