eeehmm... Sam was in fact leaving Frodo, after long and hard thinking, because he thought him dead. He was grieved and lost, but he felt the urgency of the Destruction of the Ring. Here are a few quotes of Sam:
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'Let me see now: if we're found here, or Mr. Frodo's found, and that Thing's on him, well, the Enemy will get it. And that's the end of all of us, of Lorien, of Rivendell, and the Shire and all. And there's no time to lose, or it'll be the end anyway. The war's begun, and more than likely things are all going the Enemy's way already. No chance to go back with It and get advice or permission. No, it's sit here till they come and kill me over master's body, and gets It; or take It and go.' He drew a deep breath. 'Than take It, it is!'
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In this quote he surely sees the need of their quest and the importance to go on and leave his Master.
Quote:
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'Good-bye, master, my dear!' he murmured. 'Forgive your Sam. He'll come back to this spot when the job's done - if he manages it. And then he'll not leave you again. Rest you quiet till I come; and may no foul creature come anigh you! And if the Lady could hear me and give me one wish, I would wish to come back and find you again. Good-bye!'
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In this quote he assumes Frodo to be dead and that he has to go and fullfill their task. He doesn't really hesitate to go, for he knows he must, but he really wants to come back, impossible though it may seem.
Quote:
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'If only I could have my wish,' he sighed,'to go back and find him!' Then at last he turned to the road in front and took a few steps: the heaviest and most reluctant he had ever taken.
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And in this quote he finally sets of into Mordor. That he doesn't come very far doesn't mean, he wouldn't go at all. He was planning to go, but his waiting was his fortune, for by the orcs he learned that Frodo was still alive.
So after all, I think Sam would leave Frodo there. And he would go into Mordor, and he might fullfill his quest, for he might regain power against that Land and the Ring, because of the love for his Master the Ring might not conquer him and he might be able to throw in it Mount Doom.
A new question: if Frodo was dead and Sam had done the job, would Gandalf and Gwaihir be able to find and rescue Frodo before it was all too late?