The whole idea of having Gollum swear on the One Ring was to keep the thought in his mind that if he was false to his word, then the Precious would not be his. I'm sure the Gollum part of him realized that if he stayed with the Hobbits and led them into Mordor like they asked, he had a much better chance of being able to take the One Ring. If he had merely been following them, they would have been on their guard against him more. As their guide, Gollum would have had many more opportunities to steal his Precious, which is what he was after from the beginning.
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But Melkor also was there, and he came to the house of Fëanor, and there he slew Finwë King of the Noldor before his doors, and spilled the first blood in the Blessed Realm; for Finwë alone had not fled from the horror of the Dark.
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