Strange vibrations under acceleration

[Ivo 4]

Hello,

Few months back I bought a 69 Mach 1 and I found strange vibrations under acceleration above 40mph. The vibration is strong through the steering and the gas pedal. As the car was sitting for 10 years and everything is original, I'm thinking of replacing the whole front end (upper, lower control arms, springs, idler arm, tie rod ends etc.) and the steering box.

The strange thing is when above 40mph the vibration starts, if I lift off the gas pedal, the vibration stops, but if I accelerate again, even slightly, the vibration starts again. Sometimes there is no vibration at all. The tires and wheels are new and balanced. The engine and transmission work have no issues, when I bought the car the gasket on the rear differential was fresh, so I'm thinking it was redone.

I know that there could be hundreds of issues and everything should be redone, but I need guidance where to look next.  
Thanks!

[Mach1 Driver 560]

That sounds like drive line to me: engine, trans, flex plate or clutch, driveshaft, rear end. Its new to you and all original, but do you know of any changes from stock in the drive line? Was the engine rebuilt? In the absence of information I would start by replacing the old vibration damper- harmonic balancer.

[Ivo 4]

Thanks!

I forgot to mention its 351w with FMX transition, all stock. 

Its strange, because on slow speeds there are no vibrations, even if I kick down from standing start up to 40-50mph or simply rev the engine. The FMX shifts amazingly smoothly and the driveshaft looks surprisingly fresh.

When I drive it, when the vibrations start I simply let off the gas and press it again and the shaking stops.