Do I need a slope in drain lines under sink?
You do. You want at least 1/4" per foot to keep liquids flowing and carrying any sediment or waste into your drain line. It shouldn't be hard to accomplish given your setup.
Edit
To adjust height you need to loosen the nut on the right tailpiece and raise it up on the slip joint. You may have to trim a little off the tailpiece to do that. Do the same for the lower tailpiece on the left below the inlet and above the trap. Slip joints will typically have enough play to allow you to make these type of adjustments but sometimes they require trimming.
One tip - It's often easier to make this type of adjustment if you slightly loosen all of the plastic nuts first to give you a little play when setting the slope. Just don't forget to retighten afterward and check for leaks.