It's really easy to see if the e-brake is sticking. Just put it in neutral on an incline. It should roll even if the incline is very slight.
By the way you describe it, it sounds like you could have a broken motor mount. If one collapses, there's a metal bracket that limits the engine movement, and also causes some nasty noise and vibration since the engine is basically making direct contact with the chassis.
Does it still vibrate in Reverse? If not I'd especially suspect the motor mount, since the engine would be torquing the other way. Also, pop the hood, put it in drive, and gun the engine a little with your foot firmly on the brake. Have someone stand off to the side and see if the engine moves significantly. Do this in Reverse as well. It shouldn't shift more than an inch or so.
