I am starting to get sloppy rear shifting, with the chain sometimes self-shifting and chattering like it is trying to shift when I am going at a steady speed.
I tried tightening the barrel adjustment as far as it would go, but it didn't totally eliminate the problem. Trying to avoid a visit to the LBS, is there anything else I should try first?
Most shifting problems are due to chain stretch, but before you start clean everything up. Clean the chain thoroughly, and pull the rear wheel to clean the cogs. (and old rag between cogs works fine).
With everything clean and properly lubed, shift to the smallest cog and check the tension on your rear DR cable. It should be taut, but not overly tight. Chances are you'll find it has quite a bit of slack. Screw your adjuster back in, and then loosen the cable retaining clamp and pull it finger-tight, so there's no or little slack.
With that done, you'll likely find it's pretty close to being right on. A couple of half-turns of the adjuster may be required.
Properly adjusted, the chain should "jump" quite smartly from cog to cog, with no chattering or noise.