I hate to be the bearer of bad news, but if you did not get the entire rotating assembly balanced together, you could have these problems.

If you used an 'experienced' crank, you can get away without balancing IF you make sure the weights of the pistons/rods that came off of it were close to the ones you put in it.

If you bought a new crank, I would bet my house that if it wasn't balanced it will cause the problems you describe.

When I mean balanced, I mean matched to the weights of the pistons/rods. The right way to balance is to have the entire rotating assembly balanced together (pistons/rings, rods, crank, balancer, flywheel).

Also, some new flywheels may not be factory balanced.

It was just a short block reman. Bone stock reman. Wether they threw the rotating on a hines type balancer I couldnt tell ya. I will call him tommorrow and ask.

I doubt it's the engine itself. If the crank or a balancer was out the vibration would be constant and not stop when he engages a gear. Plus the frequency of the vibration would increase with RPM.
I had a flywheel resurfaced once and the machine shop screwed it up and threw it out of balance. The vibration did not go away under load. It got worse.

Check your u-joints and the gimabl bearing when you pull it part.

When new,

2 piece rear seal 5.7's are internally balanced
1 piece rear seal 5.7's are externally balanced

Once a rebuilder get's a hold of them who knows how they do it unless you are there. At minimum, always make sure to ask.

1pc vs 2 piece crank flywheels bolt patterns are different, however dampers should all slide on the same.Make sure to ask and still check balancer, damper if weight is added on or not. Don't always trust 'what you ordered' - you have to always check part when it's in your hands. Mistakes happen.

Tech tip: when using the alignment tool, rotate engine 1/4 turn or so, insert alignment tool again, rotate again re-install alignment tool, etc, etc. This will help see if the coupler is relatively straight or really out of round.

Right thats what I thought as I had to buy a new flywheel for an older vintage 2 piece rear main crank. Now speaking from the ford side of things, we have encountered numerus bone stock 302's that were 30-40grams off balance from the factory and run smooth as can be. I just pulled the drive, will fire it up later for 5 seconds and I will know right away if it's anything related to the drive and if the vibration is still there.

update, yanked the drive, fired it up, still vibrates. Maybe got an external flywheel and it slippd past me. Time to yank it out again.

I have changed them in the boat before. If you have enough room just lift it and turn it side ways. Saves a little time

ya I got lots of room, thats what I meant by yank it out. This sucks but what are ya gonna do.