You shouldn't need to rewrap them, the product should be nearly all off the nail and anything remaining easily removed with light touch of orange stick or pad soaked with remover.
Try go thinner with base, I know you probably think its already thin but I honestly had same problem when I started. Now what I do is take brush from bottle, wipe off excess into neck of bottle and dot the brush onto each nail starting with thumb. Then go back to thumb and cover whole nail...all this without re dipping the brush back into the base coat bottle. It really is amazing how little product you need and how far it goes. Once I got into habit of doing this the residue isn't on the nails.
Try it