It would seem the pawls inside the free hub are not working properly. It's not a problem I've seen with cassettes, but quite frequently with freewheels.
I'm not sure if they can be cleaned out and relubed. If my guess is right, a new free hub shouldn't cost more than $20. (In NYC that might be $35 because of all the taxes and other charges you NY'ers are subjected to. Take it to Jersey, it might be cheaper.
Maybe another member will have an idea of the cause, and a less costly cure.
If it only happens when shifting, chances are it isn't the freehub. Save your money and skip the freehub cleaning until you know for sure that is the problem. Freewheels and freehubs are pretty maintanance free.
My guess is that your shifter isn't putting the chain onto the next chainring or sprocket. Your chain is then either floating between the sprockets or riding on top of the sprocket without the teeth setting into the chain.
Both cases will make an uncomfortable sound as you pedal; "whirrrr, clink clink whirrr...
Try forcing your shifter so that it pulls the cable and forces the chain to position correctly onto the sprockets.