Howdy and welcome!
Since your mare is registered (I assume so because you have the reg. number), you can start by calling the APHA and asking for your mare’s produce record. Of course they’ll only be able to tell you about registered Paint foals but it’s a place to start.
There’s also the All Breed Pedigree database online. I searched for a Paint horse named Shattered Dreams and one did show up, foaled in 1988. Kind of a while ago but see if the bloodlines seem familiar:
Of course there’s always the chance that the mare you’re looking for was never registered with the Paint Association and the name Shattered Dreams was just her show name or barn name.
Finally, each state has a Paint horse club. You could contact the Paint club in Nevada and see if the secretary has any records of a mare named Shattered Dreams in competitions. Good luck and congrats on your nice mare!