Scasey, while I do redo my braids one at a time which helps me wear them for over a year and they still look good, I do find baggying helps a lot.
I've got 4B hair and when I baggy, the moisture makes my hair shrink...which means even fuzz also seems to go flat against the braids so they look very neat when I take the baggy off.
To give you an example not with loose braids but extensions, notice how old and messy my cornrows look in this do:
After baggying, notice how tidy they looked (notice the shrinkage of the previously Curlformer stretched back after the baggying). I was using S Curl and I braid my hair in the back for the night before baggying, then comb it out in the AM. Basically what I did before baggying is brushed the short hairs along my hairline then baggied with Saran wrap. Nice neat hair next day.
It also works for my twists when I put them up into a style and baggy:
The next day, my twists look so much tidier due to all the fuzz shrinking back down:
The twists have no product in them. The cornrows have S Curl on them. So this trick works both w/ and w/o product. I should also mention that I am not one of those people who moisturizes her hair all the time, so that does help not mess up my braids too, I think.