When switching from circ to dpn just take one of the dpns and start knitting on it (in lieu of the other end of your circ ndl). Do 1/4 or 1/3 of your sts, depending on how many dpns you're using, then take another dpn and work another 1/4 or 1/3 of the sts, etc. Eventually all the sts will be divided between the dpns and your circ will be empty. All the sts will have been worked in the same dir as when they were on the circ.
there's only one direction I can knit since the yarn is hanging in only one place, right?
No, you can always turn and go back in the opposite dir making your RS the WS.
The 'cup' theory differentiates between whether you work your circ 'RS out' or 'RS in.' If the loop of the circ is near your body, but the ndls are away (closer to your knees), the RS of the work will be on the inside of the loop. If the ndl tips are near your body, and the loop extends out, the RS will be on the outside. If you changed this orientation when you switched to dpns that explains how your sts went from RS to WS.