You needed to add more flour....and refrigerate the dough.
Here's the recipe I use....But someone emailed me and said she thought they were tough...never had that complaint...but they are crisp!
Sugar cookies:
1 cup of butter
2 cups of sugar
Cream together and add
2 eggs beaten with 2 T milk and 1 teaspoon vanilla ( I like a bit more)
Mix with the butter mix
Add 3 cups flour mixed with 1/2 tsp salt....and as much as 1/2 cup more flour until it's stiff enough to roll out
Make flat patties, wrap and refrigerate for an hour or longer....
Dust board with flour and roll and cut...
Bake 375 for 8 to 10 minutes....I use parchment on the cookie sheets....if you don't have parchment....lightly grease the sheets.
The absence of baking powder keeps them from rising and keeps the cookies true to the shape of the cutter.....and makes them crisp

Here's mine...
3/4 cup of butter (I use hard margarine)
2/3 cup sugar
2 eggs
1 tsp vanilla
2 1/4 cup sifted flour
1 tsp baking powder
1/2 tsp salt
Cream sugar and butter, then add eggs and vanilla, then flour with powder and salt.
Chill the dough, roll a bit at time on lightly floured surface. Cut with cutter dipped in flour.
Bake 10-15 at 350.
I've also made with margarine. It's a little more difficult but works out and holds it's shape when baked. It's also a less sweet sugar cookie which I like, besides they usually get iced for those who like the sweetness.

This is the recipe that I use and frequently make up the dough ahead of time and refrigerate it until I need it but it isn't necessary and have never had a problem with them.
This is the only sugar cookie recipe I use.......no matter how many times you re-roll the dough, it never gets tough.
Great for what you are planning.
Sugar Cookies
3c flour
1/2 tsp baking powder
1/2 tsp baking soda
1 c butter or margarine
pinch of salt
Mix all together like piecrust dough cutting the butter into the dry ingredients.
Then add:
1 c sugar
2 beaten eggs
1 tsp vanilla.
Chill dough. Preheat oven to 375°, roll and cut cookies. Bake on ungreased cookie sheet for 6-8 minutes.
I always double this recipe and then frost and decorate.
RL`

I stopped trying to make rolled-out cookies until I was given this recipe by an elderly neighbor over 25 years ago. It's the only one I'll make now.

Sugar Cookies (No Stick!)

2 cups sifted flour -- divided
1 1/2 teaspoons baking powder
1/2 teaspoon salt
1/2 cup butter flavor Crisco shortening
1 cup sugar
1 egg
1 teaspoon vanilla
1 Tablespoon milk or cream

Mix and sift 1 1/2 c. flour, baking powder and salt. Cream shortening; add sugar, egg, vanilla and cream. Stir in flour mixture. Gradually add remaining flour to make very stiff dough. Can be chilled, but not necessary. Roll out on floured board and cut with floured cutters. Bake at 375° for 8-10 minutes on ungreased sheets.