There is no mechanism where you can just give up custody. You can file a Petition to Modify the custody order and ask that the children's mother take the kids. You have parental responsibilities, and so does the mother. If your situation is dire medical need, the Court will probably order her to take them, at least temporarily.
The only other alternative is to call child services and tell them you can't handle the kids. This is not a route you want to go however, unless you are completely desperate. If the kids are placed in foster care or with a guardian, you will probably still have to pay for the costs, in the form of child support to the guardian. And there will be many court hearings and you will not enjoy the process.
So your best bet is to contact the mother, and ask her to take the kids. If she won't, you might look to another relative who might be willing to take them in under a guardianship. In either case, you are going to have to go to court. An emergency guardianship can be done in a matter of days. If the mother agrees to take them, it can be done immediately and without an order. But she will probably need something from the court to enroll them when school starts, but that varies by school district.
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