did the guy build the motor himself, if he did does he know what he is doing.  Also how long ago was the rebuild done to bring the cylinders to .060 over and the new cam.  If it was not long ago did he brake the motor in correctly, did he use break in oil and change it after 300 miles. sounds like he could have done all that work and left a old oil pump in it that could not supply enough oil to the bigger motor.

I don't know the answer to any of your questions.    

But with the oil pump being a maybe....I'll start there and probably buy a new one and try it, before I rip the whole motor apart.

I would say bearing are worn out to say the least. When they start to expand they lose there clearance and lose the oil pressure. But that shouldn't cause the shake. That is something else. Sound like it needs a rebuild.

mabe a siezed oil pressure relief valve- stuck in the open position.
this would show pressure on cold start that would fade as the oil warms up.
of course this would do some bearing damage if run long.
also could affect valve operation due to loss of hydraulic lifter function- run bad.
mabe.
wrecker motors are usually guaranteed to have good compression and not smoke or leak, but you may have to put it in to find out. sometimes it pays to make them run in the yard before you buy. a good place would even help with that.
mabe again.

Thanks for all your answers fellas.  Appreciate the help.   I've been asking around....and I heard spun crank bearing a few times...and/or broken connecting rod.   Looks like I get to rebuild the motor.  Yay!   

Never rebuilt a motor before.   Never got past removing the heads.   I don't have a garage either.    However...I DO know a really skilled mechanic with lots of chevy 350's just lying around...and he's built many high hp engines before.  I know he can do it for me...but he hasn't got a lot of free time.

i built my vw motor and my roommate built his supra motor in our kitchen in our old apartment. it actually was nice cooking some dinner next to some wrenchs and such lol