First shift up after shifting down only moves as I'm releasing the lever not on both the click in and click out of the lever like the consecutive ones
Not really anything wrong there. You'll want to slowly pedal (with your hand!) while working on the derailleurs. Shifting like this simply creates more and more tension until it's unable to move the chain anymore sideways. Chains can only move so far sideways.

Also make sure the cable is free to move easily within the housing. Dropping a bit of (chain) lube at the ends and pulling it into the housing can already help a lot.