Re: Are the 60hp Mercury o/b known to shake?
This is not unusual especially on older ones. It's probably nothing to over. But...
Check your lower mounts by shaking the motor fore and aft. There should be no movement between the motor and the swivel mount. Look at where it pivots when you do this.
There are also several top mounts which can be inspected visually. They are rubber about 1 inch in diameter and almost as tall. You will typically here a rattle if one or more of these are broken. I don't know exactly how many are on yours, mine have four (older 25-35-40-50hp), two in the back, two up front on top of the brackets. My 50 has them mounted in odd, less conspicuous places.
Also, could be that the idle circuit in your carbs need a cleaning.