A Petition to Adopt will require disclosure of and notice to the biological father of your intent. A hearing or two on that will be required, most likely. Your child will likely be appointed a GAL and an investigation into your background will be done to determine his fitness as a parent.
The adoption process takes several months, and can be a wonderful experience as you start your new family.
I'd urge you two (you and your wife) to hire a lawyer and make a joint petition.
If the child has been living in Florida in more than 6 months, Texas most likely would not claim jurisdiction over the child and it would be a Florida court you would go to for a step-father adoption, who would likely allow this to take place in Florida bc that's where the child is. I suggest you find a Florida family attorney familiar with adoptions and schedule a consultation.
At least in Texas (and I assume same in Florida--ask a lawyer there), it is usually a little easier if bio-dad is willing to agree and sign a relinquishment, but possible to do even if he is not in agreement and even if he is not locatable (you serve him by publication or other alternative service), but these are things you should talk with someone knowledgeable about the relevant law in Florida, not Texas.
In general (and there are exceptions), your adoption of the child would not void any BACK child support bio-dad owes. In general (and in Texas there are is the possibility of post-termination support, you'd have to talk to a Florida attorney about Florida's rules), your adoption of the child would mean bio-dad unlikely to have to pay support in the future--you are taking on that role by adopting.
Christine Henry Andresen