Yes, "Samwise" is used in a number of places. It's mainly used in formal manner to emphasise something (by Frodo), or by suitably formal people who would use one's first name - Faramir and his men.
"I don't know how long we shall take to – to finish," said Frodo. "We
were miserably delayed in the hills. But Samwise Gamgee, my dear
hobbit – indeed, Sam my dearest hobbit, friend of friends – I do not
think we need give thought to what comes after that."
And when meeting with Faramir and his men:
"Frodo son of Drogo is my name, and with me is Samwise son of
Hamfast, a worthy hobbit in my service."
Mablung laughed. "I do not think the Captain will leave you here,
Master Samwise," he said. "But you shall see."
"Now you, Frodo and Samwise, will come with me and my guards," said
There are many other examples while Frodo and Sam are with Faramir in Ithilien.