You can give up your parental rights regardless of the reason. You can also enforce your parental rights by hiring a lawyer and filing in family court for visitation and enforcement.
You cannot just give up your parental rights. The state has an interest in having two parents responsible for the children and it just won't let you off the hook. If your husband remarries (are you divorced?) and his new wife wants to adopt the children, you can do a combined "voluntary termination of parental rights / step-parent adoption" and relinquish your rights that way.
Here's a WARNING for you: If you do not have an order in place, either a divorce decree or what we call a SAPCR, your husband can sue you for back child support. Depending on how long you've been gone, he can get as much as 4 years of back support. Even if you're barely employed or working a series of irregular jobs, this can amount to about $14,000 with interest at 6% per year.
Therefore, you need to get a divorce and have the child support issue dealt with as soon as you can.
I have a child support calculator on my web site if you want to estimate what your child support payments would be.
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