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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.
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