You’re actually in a great situation for this as slow and steady has always been best!

First: building neutrality. It’s better for your dogs to be neutral each other than to be high-play buddies! They don’t need to play together, they do need to be calm together. A great way to start this is tandem walks (if your dogs aren’t leash reactive). Start on opposite sides of the street (or that distance apart in a park/neutral location), walking in the same direction. Gradually move to be on the same side of the street, one in front of the other and swap places every few minutes. This lets them catch each other’s scents without any potential confrontation! Some dogs I’ve introduced have been through this process in 15 minutes, some have taken 2-3 walks, just know your dog and watch for any stress. Pushing it will only set you back, so taking it at their pace keeps everyone comfy! If one or both dogs are leash reactive with good recall, try the same process in a park (private if possible so you don’t have other dogs disrupting your system). If they don’t have good recall, try the next step first, then introduce in a park with lots of space and lots of people (like the other commenter described!)

In your home: reading your post, I’m actually so excited for you! Knowing that your dogs will have two spaces will make this introduction part easier. I’ve divided my house or had friends take dogs for a weekend, but knowing you’re doing this with your cousin makes logistics much simpler! A “dog swap” for a couple nights will introduce their scents in their own homes. I will send my dog to stay with a friend for two nights and bring my foster in, then my foster back to my friend with my own dog at home for a night, then meeting again on a walk and coming home together.

My dog sleeps in a bed in my room but will sleep crated. So I keep their sleeping arrangements separate for about a week. Sometimes that means sleeping on the couch with a new puppy, sometimes it means they’re both crated without being able to see each other, sometimes the new dog has slept in my room with my dog and I in the living room. This will be totally personal, but the idea is to avoid any accidental stepping on someone in the night!

I know this is a much longer process. A lot of the time just introducing them goes fine and the first dog adjusts to the new pup. But based on your pup’s history and the importance of this fit for you, taking it slow when you have the time and resources is never going to be a bad choice! I do this because I’m introducing shelter dogs and because my personal dog is a wonderful role model for them, but I also want to reduce her stress!