A.J.
OK, you are learning a couple of things FOR SURE in this trial period, which is fortunate.
1) He doesn't do anything he doesn't HAVE to do.
Therefore, sit with your lawyer and think of things you really need him to do with the kids if there are any, like certain birthdays or holidays or something and have it legally stated that he has to do them in the divorce. Keep it to a realistic minimum since he's obviously not interested and kids can sense that. Specify he must PICK UP the kids by 5pm Fridays or whatever since he's taking them as late as possible and letting you do all the work. Have it in writing somehow even if it isn't really enforceable. You don't want to leave it to him to think of doing the right thing. You don't want to ask him to follow your requests. He's already proven he won't do either. Make the court force what you need.
2) Only you can be the judge of whether it is good for your KIDS to spend a few days a month with him. Once he is away from YOU, is is good to the kids? Does he love them? Do they have fun? Is he a good role model and a good influence? Ideally, they need to know their dad loves them and wants to spend time with them and be in their lives, so the most time you can get the better. But if in reality, he doesn't love them and they don't have fun, it's better to move on and forward.
If it were me, I would go for FULL custody, and then let him see the kids when he really tried, but at least I would have as much control as possible. And I would make sure to understand he may not try, and to not bring the kids down about it. You have to keep your head in the present and your eyes on the future.
Ask your ex how many days he wants them and if he really wants them. Tell him you're willing to let him off the hook completely if he doesn't want them. Don't be antagonistic or make him feel trapped. This relationship is over, take the high road and be cool headed. If he sees he is not being forced by you, won't have to deal with you and fight anymore, and may never see his kids, he may think harder about how much and how often he really wants to see them.
Full custody can only hurt you if you don't want full custody. Fight for what you want with your lawyer.
He should be doing all the thoughtful things you mentioned, checking in with them, caring, etc. but many people don't. He isn't. You can't make him be a thoughtful person. That's what divorce court is for, to set the rules. Accept it, and know that a brighter future is ahead for you all.
I'm so sorry this is happening, blessings to you and your beautiful kids.