If you can see and feel movement in the cable, you are good. Make sure you return the shifters to neurtral -- i.e. release cable as far as you can (so low gear for front, high gear for rear) before attaching to derailleurs.

If your down tube cable stops have barrel adjusters, the in-line ones are redundant and unnecessary. I have in-line adjusters on two of my bikes only because the downtuibe stops don't accomodate adjusters. So yes, it sounds like you can ditch the in-line ones on your bike.

edited to add: Just reread the op and misunderstood the adjuster question. So no dt barrel adjusters; you just want to relocate the inline ones. You can put the in-line adjusters anywhere along the housing you want.