If you are satin stitching use a stabilizer on the back

Okay, I assumed you were doing machine appliqué. :DYou're doing hand appliqué aren't you? When doing hand appliqué, I find a backboard helpful. I have a 12 ceiling tile that allows me to pin the top of the block background down while I work on the appliqué. If I don't have an anchor, I make a big mess of my appliqué. If you have a washable glue stick, it helps to keep the applique in place on the background while you work on it too.

MY suggestion would be to not pull the thread too tight. I lay my block on the table Pin on the applique, pin on with pins going towards the center and sew it on using a blanket stitch. I do the 1/4" edge turned under and small stitches. Don't pull the thread roo tight. This is for hand appliquing.

Use fusible to position and hold the fabric applique , and use a stabilizer ( a tear away will work fine) under the layers.

Hey just remember not to use too much glue I use paper underneath the fabric and item to be appliqued... note undo the sewing you have done and then put paper under neath and slowly zig zag around the edges...
too much glue will gunk up the needle and your sewing maching. remember to change your needle after doing this because the paper will dull the point..
hugs
quiltinmama

Thank you for ALL of your suggestions, I have finished my appliqueing and no more puckers!!!!! I will post a picture of the quilt when I have the borders sewed on. You all were a very big help.