The wire size for your dryer should be #10. Or, 10/3 on a 30 amp breaker. The dryer operates on 240 volts and the DW runs on 120 volts.
So you have to change the breaker in the panel to a 20 amp single pole breaker. Land the white on the neutral bus, abandon either the black or red, connect the ground to the ground bus bar and connect the remaining black or red to the single pole breaker. This should all be done already except for the extra wire you do not need. Just cap it off. The
DW unless provided with a cord and plug, requires a disconnect.
See if you can get a listed cord and plug that is for the DW. That way you can just install a receptacle and be done with it.
If you cannot get a listed cord and plug, you need the disconnect, and hard wire from there. It's okay to use #10 wire on a 20 amp circuit. Hope that helps.