I think the best solution is to completely divorce damage and utility cards. Make them separate decks with separate draw CDs. If neither one comes with a cost to the other then optimal play will require using both.

I'd change damage Draw to have a 30 sec CD and provide one of each seal when used out of combat. Add utility Draw with a 30 or 60 sec CD. Each deck gets its own Play button. Remove Sleeve Draw or change it to immediately draw one damage card and one utility card. Both decks would have their own Redraw, possibly sharing charges.