BobP, sorry you are going through this. Unfortunately this is typical WW behavior.
I say this a lot, something that was said to me in my sitch when my W wanted to move out and get her own apartment (it is specific to women but it is really gender neutral: "Women do not need their own place to find themselves or to work on the marriage. Women need their own place to sleep with other people."
As AS said, her moving OM in, even if he isn't sleeping there, is typical. And while it was her house, you are married and a resident of that abode. Most lawyers would have told you that you should stay....and that you had a legal right to stay. I really wish you had consulted a lawyer before making such a big decision. The end of a marriage is a largely a legal proceeding. You need legal advice to guide you through that.
As far as the agreement on dog care. Terrible. Your justification for moving out was to give her space. How does going to the house so often accomplish that? I would rearrange it. When dog is in her care it's her responsibility. If she cannot take care of the dog, then you should take the dog 100% of the time! Look at how badly seeing what you've seen has already set you back?
You are making a lot of mistakes. I don't mean to beat you over the head but sometimes bluntness is necessary, and I would love to see you start taking back some respect, and stand up for yourself.
"Sorry, the current arrangement with the dog is not working. I feel I should take the dog full time and you can have visitation rights. Unless you think you can handle the care of the dog for the 2 weeks at a time it is with you?"
For dog drop offs and pickups, it is just like a child drop off and pickups. Picking the dog up from you is her responsibility (IE you do not deliver the dog to her.) When it is your turn to pick the dog up from her, you pull up in the driveway, or curbside and the dog is to be brought out to you.
She doesn't get to have you move out, do whatever she wants with whomever she wants, but still give you access to the place because it is convenient to her. Waywards are the worst because they will take and take and take until you put your foot down. So give up the keys to the house. And do not agree to take care of things for her. She fired you as her husband, she doesn't get to have her cake and eat it too!
Oh, and her manipulation attempts are disgusting to me. Snuggling up to you, being sad you are leaving. Etc. It was all manipulations to get you to be "peaceful" and to be there when she needs you. Rip the bandaid off. She doesn't get to have you when she needs you, and then do whatever she wants with whomever she wants. She is not respecting you, at all. Start doing things that command respect!!
M(53), W(54),D(19) M-23, T-25 Bomb Drop - Dec.23, 2017 Ring and Piecing since March 2018