That first issue with the shifter not "catching" sounds like it might be gummed up with old lubricant. It's a known issue with Shimano shifters, I've seen plenty about it on here and experienced it on several occasions, flush the shifter out with WD-40 or GT-85 or something and see if that helps. It certainly won't hurt.
As for the barrel adjuster, think about it for a minute. When you unscrew the adjuster from the shifter housing, you're effectively lengthening the cable housing slightly by doing it, which has the same effect as shortening the cable - it tightens the cable.