I'm not sure if this will answer your question, but let's look at what we know about The Oracle from the sequels.
SPOILERS BELOW
First, we know that she was created to understand the human psyche. So her predictions aren't necessarily "fate". If a scientist uses a theory to accurately predict the outcome of an experiment, it's not fate, but science. I believe that The Oracle's ability is of this nature.
Second, such prediction doesn't necessarily mean they have no free will. This is a classic philosophical dilemma that has had much ink spilled over it. It would take far too much time to go over here.
A third point to bring up is when Morpheus says "there's a difference between knowing the path and walking the path." This, perhaps, is is the part where Morpheus' desire to free will comes into play. He knows the path (predicted), but walking the path exercises his free will.
Finally, we have this excellent quote from The Oracle which I believe speaks for itself. "Because you didn't come here to make the choice. You already made it. You're here to try to understand why you made it."
But Morpheus and Neo don't know she's one of the Machines. So, why did they go to her? I've only got speculation, but I speculate that Morpheus went to her for help and was probably desperate for an edge in the war against the Machines. Neo, on the other hand, is a bit different. He doesn't necessarily buy her story. She says it of him herself "as soon as you walk out that door, you're going to remember that you don't believe in any of this fate crap. You're in charge of your own life." Neo later perhaps starts to believe her when he realizes they have to kill Morpheus in order to protect Zion "The Oracle said this would happen." I'm not sure whether his mission to save Morpheus was a result of his wanting to defy fate or his belief in what the Oracle said that he would be able to save Morpheus, but at the cost of his own life.
I hope this gives you some information even if it doesn't fully answer your question.
