Evinrude 1986 60HP 3 cyl

Now What on the 60hp

when it shuts off does it sound like it starves for fuel, or like a quick shut off like you grounded the ignition (hit the kill switch)? Try to eliminate the area that it might be to help speed up the problem finding, you know it has compression, so we need to find if it is spark or fuel. (i ask about spark because you could have a tempremental switch or loose wire)

Does it keep running if you bring the throotle back to low position after trying to go to high position?

A couple of things to try.
Use an inductive timing light and check all plug wires for spark when it boggs down.
Use a squirt bottle with fuel mixture in it and squirt fuel into each carb when it boggs down.
This should tell you if you have a fuel or ignition problem, compression is OK isn't it? Are you using fuel stabilzer in between runs. Sounds like it sits for a good period of time before you use it. Could be gas is going sour as it sits. You may have to pull carbs again and clean! It may be the floats are sticking, causing it to run rich. If the compression test looks good and the spark tests good,the spark plugs all look the same then maybe its time for the carbs to be gone through. If you REALLY don't want to open up the carbs, get yourself a can of Seafoam and spray it into the carbs while the motor is running. You might get lucky and the Seafoam will dislodge/dissolve the obstruction(assuming that you have an obstruction). Put the rest of the Seafoam in your fuel tank.