The lawyer may tell you what typically happens in such cases. That's okay, you need to know that. But will he have
strategies to get the best outcome for the children? Court won't care much if at all about you, it will claim it's about the children but the reality is that mother may get consideration too. Don't pay a retainer until you're satisfied this is the lawyer for you.
At first I thought she was seeing the children only every other weekend, basically an EOW parent, a stereotype often expected for fathers. But if she is caring for the kids while you work and you care for them the rest of the time except for a couple overnight every couple weeks, then that is not so clear-cut, she sees them somewhat frequently. You need to ask the lawyer(s) whether the court and agencies look only at overnights or also consider the frequency or percentage of time with the children.
Since they'll be in school soon, do you plan to start using daycare before and after school? That would also result in fewer exchanges with her.
Also, while you're handling all this legal stuff, keep in mind that the IRS wants it clear who gets to claim the children for tax deductions, etc. So review IRS Form 8332 and
related articles.
I think you made a comment previously about going on disability or something like that if things don't go your way. Ponder whether taking such paths would impact the court's view of you as a capable parent, also whether that would incline the court to think you're more concerned about the money than the children, a stereotype fathers can unfairly face.