"At one point in time they were still in contact with family members," he said.
White said tips led investigators first to Florida, then to the Orlando hotel where the couple was staying, and ultimately to Universal Studios, where they were arrested after watching a parade.
Law enforcement officials spent "countless hours" trying to find Linscott and Dow, who are from Plaistow, N.H., a town of about 8,000 residents in Rockingham County on the Massachusetts state line, marshals said.
The Orange County Jail in Florida, where they are being held in protective custody, said the couple had declined all interview requests.
Dow's grandmother, Helen Fram, who lives next to the couple in a two-family home in Plaistow, said Thursday that she's certain her grandson and Linscott did not hurt James. She said the boy sometimes banged his own head on the bathtub or threw himself off the toilet.
"I can hear anything if anything is going on," she said. "They would never, ever hurt a baby. Never."
She said they initially went to the hospital with James but got scared when they found out police wanted to talk to them.
"They figured they didn't do anything, so they took off," she said.
Fram described her grandson as a loner and said he sometimes did siding or roofing work, though he is unemployed right now. She said she did not hear from him while he was on the run.
Asked why they would have gone to a theme park, she said that was one thing she did not understand.
In the small downtown of Plaistow, Joe's Barber Shop owner Dan Diemand said every third or fourth customer for the past week has wanted to talk about the case, wondering if Dow and Linscott had been caught. A lot were angry.
"Especially parents, I think they take it very personally when a child is hurt," he said.
Tuohy reported from Concord, N.H.