You would need his consent for a stepparent adoption or you would need to terminate his parental rights for abandonment. The longer he is gone and not seeing your son or supporting him the better your case.
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You should consult with a family law attorney in your area immediately. It is probably prudent to initiate an action against the ex boyfriend in Florida now, while he is still a resident. Once the Court ha jurisdiction over him you will be able to require updated address information. Additionally, if he is arrested for the Ohio warrants - he'll be easier to find if incarcerated.
In any event - you should consult with a family law lawyer before ex boyfriend leaves Florida for parts unknown.
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It's impossible to estimate the odds of him getting visitation that being said my colleague is correct it would be prudent to begin proceedings against him now while he easier to locate. My firm handles such matters and offers free case consultations. Our number is 813 830 2361. Thanks
This answer does not, nor is it intended to, create an attorney-client relationship or constitute attorney advertising. Rather, it is offered solely for informational purposes. The facts of each case are different and unique, it is critical to consult with qualified counsel with whom information can be shared and assessed under attorney-client privilege, so that competent and quality advice can be obtained on which you can make informed decisions