Paul is giving his body over to the Word.
To understand how Paul means by “giving his body over” we look at the big picture of what Jesus is teaching. Forgiveness. Now this forgiveness can be learned from The Parable of the Shrewd Manager.
There was the word whose disciple was accused of losing the Truths being told him. So he called him in and asked him, ‘What is this I hear about you? Give an account of your "Listening", because you cannot be the "Listener" any longer.’
“The disciple said to himself, ‘What shall I do now? The speaker is taking away my job. I’m not strong enough to dig, and I’m ashamed to beg. I know what I’ll do so that, when I lose my job here, people will welcome me into their heart.’
“So he called in each one of the Word’s debtors. He asked the first, ‘How much truth do you owe the Word?’ “‘Nine hundred gallons of truth,’ he replied. “The disciple told him, ‘Take your bill, sit down quickly, and make it four hundred and fifty.’ “Then he asked the second, ‘And how much do you owe?’ “‘A thousand bushels of truth,’ he replied. “He told him, ‘Take your bill and make it eight hundred.’ “The master commended the dishonest disciple because he had acted shrewdly. For the people of this world are more shrewd in dealing with their own kind than are the people of the light. I say to you, make friends for yourselves by what they say that is wrong, that when it fails they might recieve you into eternal dwellings.
“Whoever can be trusted with very little can also be trusted with much, and whoever is dishonest with very little will also be dishonest with much. Therefore if you have not given memory to the incorrect things that have been said, who will entrust to you the Real Truth? And if you have not given memory to what has been given to you by another who will give memory with what Truth you give?
No one is able to serve two masters either he will hate what is being said and love what he is about to say, or he will be devoted to the word and despise his disrupting thoughts. You are not able to listen to the word and be prepared to say something. (Luke 16:1-13 Decoded)
They define καυθήσωμαι by the root καίω meaning "to burn" but can also means "to Light".
ἔχω also means "hold" besides the usually translated "have".
ὑπάρχοντά also means "already have" besides the usually translated "possessions".
And if I give away all of the thoughts I already have and if I deliver up my body that I might light the fire of Love, but do not hold on to the words, I profit nothing. (1 Corinthians 13:3 Decoded)