It would require replacing your brake cable housing (and you may want to replace the cable at the same time) but there are brake cables with inline barrel adjusters. In these, the barrel adjuster is usually situated somewhere between the brake lever, and the braze-on at the front of the bike. They look something like this and are sold by themselves, or often included with the cable housing. As your local bike shop and they should be able to get you something that will work.