Is the sign for 'dish' the same as 'plate'?
Yes, in ASL, the signs for 'dish' and 'plate' are exactly the same. The context of your sentence will naturally let the other person know which English word you mean.
How do I sign 'washing dishes'?
You can sign 'wash' and then 'dishes'. Alternatively, there is a specific, natural action for 'washing dishes' where you simply mimic scrubbing a plate with a circular motion using your dominant hand over your non-dominant hand.
Should my hands touch at the beginning of the sign?
Your hands can lightly touch or just start very close together before pulling apart to outline the dish. The most important part is the outward shaping movement, not whether they make contact.