As I recall from seeing Ashtray's car, a previous owner disconnected the ported vacuum switch circuit that is used on the 68/350. They rewired the vacuum lines without passing through this switch. I didn't pay much attention, but I think they buggered up the switch too.
Who knows if they rerouted the hoses to suitable places? Check to see if they make sense to you. This switch routed vacuum around depending on engine temperature. It's acting as though your stuck with late timing because this switch is missing. That will make the car drive like a stone.
I haven't been able to find a clear drawing for this anywhere. Maybe someone can post a clear photo? I'm sure I saw one somewhere, but I can't find it tonight.
I think with this thermostatic vacuum switch system ripped out, the factory distributor advance curves are out the window. You may get it working okay without this stuff by configuring it for the flow outlined in black for "off idle or cruise position, timing advanced". Where is the other side (the linkage side) of the distributor diaphram connected?
As long as it's Ashtray's money we're spending, Ames sells this vacuum switch for $37 and the hose kit for $56. Starting with the correct OEM configuration may be the best starting place.
You say the car developed hesitation. Did the problem start after a period of working okay, or is is something that was noticed after making timing adjustments? Was this hesitation after setting advance "to stock" with this wrong vacuum system in place? The factory procedure probably won't arrive at correct timing with stuff hooked wrong. Maybe it needs some more advance the way it's hooked up. Crank the advance up till she pings and back off a tad. Or disconnect and plug the hose to the rear port of the distributor advance can.
Well, just some ideas to kick around. If someone's put a single port distributor in, ignore what I've said.