I'd have to agree with JohnMC about the center support bearing. There is a small spring that sits within the rubber mount to help hold the driveline and absorb some of the shock. The metal clicking noise you described is probably the spring coming out of the mount and rubbing on the edge of the driveshaft. If the driveshaft isn't properly supported anymore, you will get a lot of vibration while moving. It only seems to get better when going faster, but the truth is, don't wait on checking this out. This is what happened to me, and since it took me a while to figure out what it was(because it only happens while you are moving), I ended up having to replace one u-joint as well as the center support bearing. I'm actually lucky the front half of my driveshaft stayed attached.
Phil
69 142S

A broken motor mount isn't really a source of massive vibration - it will just let the normal slight vibration transfer to the car better - but its nothing that would scare anyone. And in any case the engine wasn't vibrating at a rest so it seems to rule that out plus any other engine problems.

My first thought was perhaps a u-joint or drive shaft center support bearing - due to the continued vibration when rolling with the engine off and the groaning on accelleration, but what steers me away from that is that you reported feeling vibration through the steering. That seems to indicate something loose on the front suspension or steering linkage - possible giving one of the front wheels enough slack to chatter (although I'm not sure that would be related to acceleration or not?).

What I would do is:
1) Jack the whole front end of the car up. Spin the steering wheel back and forth and listen for strange noises. Wiggle each front wheel top and bottom, left and right. Look for slack, take the tires off and look for damage. Could be a (really) bad tie rod, idler arm, wheel bearing, suspension bushing (although I've never heard of one suddenly going bad). Spin each wheel and listen for rumbling or scraping sounds.
2) Look carefully at the u-joints and center support bearing as well (other than the steering wheel vibration this still sound like the main culprit). If one went out that dramatically there should be some visible evidence like little chips of metal or obvious slackness when wiggled.

Good Luck and let us know what happens.

If it is coming from the front, I doubt that it is any of those things.
Maybe a front wheel bearing came apart, a brake pad let go, maybe even a
weird tire or rim problem. Driveline problems range from under your seat
to the back of the car, and tranny problems are where the tranny is,
still not in the front of the car. If it was a serious motor mount
problem it would be noticeable at shift points.
--
George Downs Bartlesville, Oklahoma