Try this

When you say the vibration is "shaking" your boat it sounds to me like it is your engine running rough and not an internal vibration.... And with the other symtoms you discribed I would try this.

If your model # E70ELCSA

I looked at the carb diagram for that engine and it appears there is no Idle mixture adjustment on those carbs. But there is an idle orfice. Go to Advanced Auto Parts and get a can of "Sea Foam" it is great stuff I use it in every engine I have, from lawn mowers, and tractors to my Durango. Then go to this link and find part #25, 26 & 27.

url=http://www.boats.net/parts/search/BRP/EVINRUDE/1980/E70ELCSA/CARBURETOR/parts.html

Pull those parts off the carbs, but be careful not to mix them up, clean them with some Sea Foam, you can even squirt some into the holes if you have a squirt bottle. There are two more orfices if you want to clean them also. they are part #35, 36 & 37. And #1, 3 & 33. Just be careful, and replace any damaged O-rings and gaskets.

Then read the can and add the correct amount to about 6 gallons of "NEW fuel", you CAN mix it a little strong, then start it and run it until that fuel is gone and see if there is any improvement.

Let me know how you make out.

I agree with JabIII, however change your spark plugs as well.