No, it's using a shield and weapon that causes problems. If you have a shield in one hand, you can leave the other hand free or hold a focus/interact with a component pouch in it. Technically if you're casting a spell that has no material component you have to drop, then pick up the focus as RAW you can only use the hand holding a focus for somatic components if the spell has a material component, but I've never seen anyone enforce that, and the 'tie a string to your wand, drop it, pick it up' sequence is just silly, and can be completely bypassed by using a material component pouch or putting a crystal focus on a necklace.
You do need to be proficient with the shield to cast while holding it, though, and it's a full action to remove a shield, unlike dropping a weapon it's not a free action.
Dragon's Breath can cause absolute havoc when cast on a familiar. 3d6 elemental damage of your choice, for the fair price of your concentration, a second level spell slot, and (most importantly) no party members actions.