Let your dishes air dry; if you don't have an automatic air-dry switch, turn off the control knob after the final rinse and prop the door open a little so the dishes will dry faster.
As with hand planes, the key to using chisels successfully is keeping them sharp. If they are not kept sharp, your work will suffer. If you are wary of taking your good chisels to the stone, consider buying a cheaper set to practice on. Store your chisels boxed, in a tool roll, or in a wall rack. Avoid magnetic tool racks; when chisels are magnetized small metal filings will cling to their edges, making honing more difficult and dulling a keen edge much faster.