Tighten the cable at the brake itself, you might need a second person to help you with that. In general the LBS is right, but these do add some convenience. Plus make sure to install the ferrules correctly...
If I am understanding you correctly, that barrel adjuster as you call it is basically a cable housing stop if you remove it the housing will go into the hole in the lever. You at the very least need the cheap fat rounded cable stop for cheap road bike levers found in all purpose bicycle cable packs like found at Walmart. look at the picture in this link and you will see the part i am talking about, kind of looks like an end cap but more fat and round piece of alloy.
https://www.acehardware.com/departments/outdoor-living/sporting-goods/bicycle-parts-and-accessories/8305336?x429=true&utm_source=google&utm_medium=cpc&gclid=EAIaIQobChMIpe6DuerT5gIVj8DACh3tZALlEAQYAiABEgJWnPD_BwE&gclsrc=aw.ds
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