I assure you, the actors notice. Shakespeare is always funniest in his tragedies. The comedies have almost universally aged badly, but the kind of humor he puts in the tragedies holds up much better.

A piece of advice I was given very early on is to find the funny parts of the tragedies, and that has served me well. Hamlet in particular is very clever about wordplay -- far better than nearly all of Shakespeare's actual clowns (though the clown role in Hamlet, as the gravediggers, is among his very best comic writing).