How do I get sugar cookies to stay in shape?

Assuming you're using a recipe for rolled out sugar cookies, you may not be rolling the dough thin enough. I roll the dough to less than 1/4" thick and then I don't have the problem of the cut outs losing their shapes.

This happened to me, and I actually figured out that if I make my oven 25 degrees higher than what it says in the recipe, then the cookies turn out fine.

your oven may not be hot enough.

If the cookie dough melts first before it bakes, that will cause the blobs.

Get an oven thermometer and check to see if your thermostat is accurate.

Find a recipe specifically for rolled out, cut out cookies. Try allrecipes.com.

Cut out the cookies and leave them in the refrigerator overnight before you cook them. Make sure the oven is fully preheated before putting the cookies in.