The bombs were hidden in plant pots spaced 50m (164 ft) apart, and detonated by mobile phones within half an hour of each other, they add.
Twin bombings are a regular occurrence in Thailand's three southern provinces, which are wracked by an insurgency.
But it is rare to find such attacks in Thailand's tourist areas.
Local reports say the woman who died was running a food stall in the area of the blasts.
Briton Mark Gainsford told the BBC he had been in a bar near the blasts: "I heard people shouting 'Bomb! Bomb!' but I didn't hear any blast.
"I ran out to see if I could help. I saw eight to 10 people injured, on the floor. The police arrived very quickly."
Hua Hin, an upmarket coastal resort on the Gulf of Thailand, is popular with foreign and Thai visitors.
Its royal palace is famous as the preferred home of King Bhumibol Adulyade.