The purpose of a barrel adjuster for a front derailleur such as this one (non-indexed) is only to remove slack while the derailleur is in its resting position. That slack is harmless. If it bothers you, detach the cable, screw in the inner limit screw slightly, reattach the cable, and then back out the limit screw to where it was. This will use the return spring to take up the slack. But it works only if you succeed in getting the front derailleur in the proper resting position, so it may take trial and error.