First of all - WELCOME!!! Congratulations on your little one!!!!
Sorry to hear you are having trouble
So she is only sleeping from 3:00 am - 8:00 am, and then a few hours during the day?
Wow that must be frustrating... that certainly isn't enough sleep for a 5 week old. It could be the constipation that is making her fussy. Constipation is common in young babies - there are several things you can try to help this. This is my favorite resource for constipation information
constipation, infant
Here is some great information on milk allergies in babies.
Milk Allergy in Infants
Milk allergies are often difficult to work with, as babies who are allergic to milk proteins are often allergic to soy proteins as well. You can also try a hypoallergenic formula like Nutrimigen or Allimentum.
She could also have reflux - which I only suggest because she was spitting up. While the change in formula could have resolved the spit up, she could still be having discomfort from the acid reflux in her esophagus. I would ask your pediatrician about it - there are formulas out there to help with that too - Enfamil AR, Similac Sensitive RS - and medications like Prevacid and Zantac.