Leaving baby overnight?

When did you first leave your baby overnight? My mom (grandma) is someone I trust 110% with our baby. She actually came and stayed with us for a week after delivery to help me out, and I honestly think the baby might be more comfortable with her than my husband. We are first time parents, and he hasn't necessarily been a bad baby, but we have had a lot of trouble with gas and tummy issues. And the max he ever sleeps is two hours. Usually only one. So we are majorly feeling the exhaustion. I've also suffered from the baby blues. Our LO is five weeks old, and only on formula. My mom thinks it would be a good idea to just give us one night to catch back up on sleep, but I just don't know. Feel like I'd feel like a horrible mom. Even though she's insisting.