From experience, I've come up with two solutions. And no... it does not mean banishing my baby to another room.
Why don't you try this.
Keep a scotch tape dispenser or a lint remover roll next to you at all times. Then all you have to do is regularly go across the top of your project area (especially the part where you're stitching) with the sticky papers--- and voila! It picks up your pet hairs.
IF you've already stitched hair into your projects... you can try pulling out strands of hair using tweezers. So yeah-- you'll need a magnifying glass and tweezers.
Maybe it may be too late (and you don't have the patience to go through your giant projects anymore). All you gotta do is think this way.... you're preserving your pets DNA for the future. hahaha.
At least this way, in future projects, you can prepare the lint remover or scotch tape when you stitch a new project.
Hope it helps! :)