Hello Kayjaybee90 and welcome to the forum from South Carolina! I wouldn't worry about the threads running as DMC threads rarely if ever run. I wouldn't say it is impossible but very improbably. I have made a lot of afghans using DMC threads and they have been washed in a washing machine and tossed in a dryer numerous times and have never run. For the last 5 or 6 years I have mostly stitched large projects and have used washable fabric markers to grid the fabrics (draw the chart lines directly on the fabric) and have washed those pieces to remove the ink and have never had floss run. It can be different if you are using hand dyed specialty threads, but with most major brands of cotton floss, you should be safe.
I hope that you will stay with us and show us the sampler you are making. We love pictures and we especially love baby samplers and Winnie the Pooh baby samplers are among the favorites of our members.