If it's the thing I'm thinking of on the back of the switch where the wire comes out on the inboard side of the brake lever, it's not an adjuster. It's a retaining nut that holds the switch in place, so should be done up as far as it goes to hold the switch right back. If it were loose, it would make the switch less sensitive, but yours is over-sensitive, so isn't the problem.
There's additional adjusters for the brake cable. One is where the cable comes out of the lever and the other is at the brake caliper. These take up the slack in the cable to make the lever operate the brake at the right point, but as the brakes wear, they usually max-out, so you have to shorten the cable where it clamps to the caliper. A properly adjusted cable should help.
The brake lever should have a spring in it that makes it return regardless of brake adjustment, so if it isn't fully returning, you have a stiff cable or lever, or even possibly a broken spring. If you haven't already done it, make sure that the cable and lever pivot are well lubricated. Check that the cable hasn't frayed a bit. A new cable is only £1.50 in Wilkinsons, so try one of those if all else fails.