Instead I wound up getting a little bit crafty. Last week, Dogs or Dollars mentioned a tutorial about making your own bows. So I dragged out a bunch of old magazine pages I had saved from grade school (I'm a sort-of-recovering pack rat...) and got to work.
I slightly changed the tutorial because of the size of my magazine pages. I used 3 10.75" strips, 3 9.75" strips, and 2 8.5" strips, all 0.75" wide, plus one 3.5"x1" strip. And I put the bows together with a glue stick, instead of double-stick tape. I think if I do this again, I either need to use more glue or a different kind of glue, since a lot of them fell apart later...
I was only able to wrap a few presents in comics pages - it might have worked better if I had told anyone at home about this a few weeks ago. Instead I just had two Sunday funnies and one weekday comics page. So I also wrapped presents in brown paper bags and put one in a box.
Looking through those old magazine pages, I found a bunch of animal pictures, so I decided my young male cousins didn't need bows. The Star Wars-obsessed 10-year-old got everyone's favorite 'droids :)
All in all, I'm pleased. Tape is still involved in the wrapping, so it's not totally plastic free, but basically everything else involved was destined for the recycling bin. The gift tags were made from a piece of pastel paper I've also had laying around since high school. I've already given some of the comics+magazine bow presents, and they didn't get many comments, so I guess that means no one thought they were weird?