Resenting your kids being around is a red flag. Coupled with his negativity and comments about living in a car or being dead, his behavior seems manipulative.
If he's not creating a generative, positive space in your now-shared home and is trying to push your kids away, he's not in a position to be a good boyfriend.
You can help him think through his options, but you're not stuck just because he's in a bad place in his life.
It sounds like he needs to do some work on himself before he's ready to be in a relationship with you.
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