Help me assemble a kitchen sink drain
A tailpiece   is the solution if the top of the piece in your photo is a slip connection. You may have to get one that has a slip connection on top to accept the tail piece you buy.
If the cut tail piece is to short to work with you can  exchange the long IKEA type basket (you called it a sink drain)  for a more traditional basket, they do not extend down as much.
Then cut down a 6&quot; tail piece to give you the distance you need for the dishwasher tail piece.