Help!! My easter cookies lose their shape when I bake them!?

I am making some cookie cutter cookies for easter, but they lose their shape after I bake them! My bunnies and eggs are turning into golden-brown blobs instead of the cute little shapes that I wanted.
Am I doing anything wrong? How are people able to make those cute little holiday cookies??
Do you think it would help to use actual butter instead of margarine, as well?

How can I make these types of Easter cookies (and keep them fresh for shipping)?

I have a link to the picture:
http://noigiftsandgourmet.com/cookies_lewis/images/d_easter/lg/smalleggs.jpg

I want to make these types of cookies, the frosting, the shape. All of that.
Also, how could I make sure they stay fresh if I’m shipping them across the country?
Thank you!

Gizzi Erskine makes Easter cookies

This video features Gizzi Erskine whipping up a classic holiday favourite of Easter cookies with the help of the unrefined sugar specialists – Billingtons and Bakingmad.com. So round up the kids, get your apron on and try out some of the great recipes from bakingmad.com. Ingredients 90g Butter, softened 100g Billingtons Golden Caster Sugar 1 Egg ½ tsp Vanilla Extract 200g Plain Flour ½ tsp Baking Powder ½ Salt Makes about 20 cookie cutters depending on the size of the cutter -Sieve the flour, baking powder and salt into a bowl and set aside -In a large bowl cream the butter and sugar until pale -Beat in the egg and vanilla extract -Add the flour to the butter mix and mix until the dough is formed -Wrap the cookie dough in cling and chill for at least one hour -Preheat the oven to 170 C/fan 150 C -Roll the dough out on a lightly floured surface to about 0.5cm -Cut out the cookies and transfer to baking sheets -Roll up the trimmings and repeat stages above to make more cookies -Bake for 8-12 minutes until golden -Remove and transfer to a cooling wire and leave until completely cool before decorating.