I decided to continue stitching with needle and thread on the backside. The strands of yarn don't come loose anymore, not even with any amount of tugging. Those corners will also get sewn shut so they'll be flat and stick up, so my thread work won't ever really get disturbed at all.
Thank you everyone for your advice.
When I get it all done I'll try and take a nice, clear picture of it. I'm so proud of my first knitting project (and that I actually followed through and finished it!), and I really hope the person it's for loves it. <3