single mother has full custody of child under FL state law can I move to another state without father's permission

DCF is required to investigate the calls it receives as long as the actions described in the calls seem to be child abuse so there is not much you can do to stop an investigation. If you have sole parental responsibility, you can live wherever you want. If you are the primary residential parent and have shared parental responsibility, there is a procedure to follow to relocate. Only the father is involved in the process. Other family member do not have input into this decision.

In reviewing your question, since custody was established by prior court order for child support with the child's residence established at a specific address here in Florida, it appears that you must first comply with the Florida's recently enacted relocation statute.

Florida's relocation statute - F. S. 61.13001 titled Parental Relocation with a Child (enacted in 2006) - restricts the right of the custodial parent to move away to a principal residence
more than 50 miles away from his or her principal place of residence at the time of the
entry of the last order establishing or modifying the designation of the primary residential
parent or the custody of the minor child, unless the move places the principal residence of
the minor child less than 50 miles from the nonresidential parent.

Under the new law, a parent who receives custody and wants to relocate must now file and serve a written notice upon the other parent and prove, by detailed information, why relocation should be granted. The non-custodial parent now has the right to file a written response within 30 days as well as having the right to an evidentiary hearing. The Notice of Intent to Relocate must be served upon the other parent and all other persons who have court-ordered custody and/or visitation rights. The Notice must include the following information:
The new law requires a custodial parent obtain a court order authorizing relocation if the move will be more than 50 miles from the non-custodial parent. If the parents agree on the move, written agreement may be filed with the court confirming consent, defining visitation rights for the non-relocating parent and any other persons entitled to visitation and describing, if necessary, transportation arrangements related to visitation.
If there is a judgment, pending court case or decree of record pertaining to the child’s residence or visitation, the parties are required to file the agreement with the court and obtain a court order. No evidentiary hearing will be held unless one of the parties requests a hearing within 10 days after the agreement is filed with the court.
If the parties do not agree, the statute has a detailed procedure to be followed by the custodial parent wishing to relocate. First, the parent seeking to relocate must prepare a Notice of Intent to Relocate. The notice must be signed under oath under penalty of perjury and include:
• Address of the new residence;
• Mailing address of the new residence, if known;
• Home telephone number of the new residence, if known;
• Date of intended move;
• A detailed statement of specific reasons for proposed relocation and, if based upon a written job offer, the written job offer must be attached;
• A proposed post relocation visitation schedule and transportation arrangements;
• And a warning in large type that the person receiving the notice must file an objection with the court within 30 days after service of the notice.
The Notice must be served on the other parent and every other person entitled to visitation. If there is a court action pending regarding the child, service of process may be according to court rule, otherwise, service must be same as in filing and serving of an initial lawsuit or via certified mail restricted delivery, return receipt requested.
If no objection is filed within 30 days, it is presumed that relocation is in the best interests of the child, that relocation should be allowed and the court is to enter an order authorizing relocation and adopting the visitation schedule and transportation arrangements contained in the notice. If an objection is filed, a hearing must be held.
The court can issue temporary orders restraining relocation or authorizing relocation after considering evidence presented at a preliminary hearing. The statute now has eleven factors to be considered by the court:
• Nature, quality, extent of involvement and duration of the child’s relationship with the parent proposing relocatio