Hi Jen,
Im so sorry to hear about Becky

I hope her remaining days can be as easy a possible for her and that she doesn't feel any pain.
I don't think you should feel like you are betraying her, I'm sure she wouldn't want you to be alone.
I'm sure you are just focusing on Becky for now and spending as much time as possible with her so she know's you love her!
Only you can know the right time and when you'll be ready for another pup. Just be prepared for a massive change, I think we all too easily forget the puppy days
Welcome to the forum