How does a spider's legs not get caught in silk when wrapping prey?
There are two parts to this answer.
Spider Legs
First, I would like to mention that spiders avoid sticking to their webs by more means than just non-sticky anchor strands to walk on.  To help keep the glue from sticking to their legs, an oily substance apparently covers spider legs.  Furthermore, the same study found that:

In addition, Eberhard and Briceno also noted that the spider’s gait seemed specially designed, such that it “reduced the area of contact between the sticky droplets and the setae [spiny leg hairs].” This, combined with the fact that the thread touched many (rather than only a few) branching hairs, made it very unlikely that any glue would slide all the way down the hairs to make contact with the main part of the leg – which would have made permanent adherence more likely.

Spider Wrapping Silk Fiber is Different
The main reason spiders do not get stuck to their silk when wrapping prey is because wrapping silks are different from silks used to make webbing.  There there are seven known types of silk that spiders can produce.  Aciniform, or wrapping silks are the strongest silk from a spider and specifically designed for wrapping.  This silk is typically dry, and does not contain the high amounts of N-acetylgalactosamine which give the distinctive quality of being sticky.   For a very detailed study on wrapping silk architecture, please see this recent study from 2015.