Philosophy was my major, so I've had a lot of discussions about this very topic. The question you raised is a good one: "If we think we have free will, does that mean we have it?"
While I believe we have free will, one could easily argue that feeling you have free will doesn't necessarily mean we have it.
Of course, where things get tricky is when you factor in genetics and predispositions. Are you familiar with the philosophical story of "Chanelle, Sabrina, and the Oboe" by Bruce Waller?
In the story, two people Chanelle and Sabrina have a discussion about free will. Chanelle argues that even though she has a gene that causes her to like playing the oboe, Sabrina is still freely choosing to play the oboe. Sabrina argues in order to have freedom, she can't be influenced by any natural/biological causes or factors.
I think when it comes to free will, the soft-deterministic view is what's best and most logical. In order to have free will in the way that Sabrina describes, one must not be influenced at all by external forces. Of course, the problem is that under this definition of free will, one cannot be considered free because their “choices” are actually random and not influenced by anything. Therein lies the paradox. Being truly free means being your actions are random and unattached from anything. That’s not acting free, that’s acting crazy.
In the soft-deterministic view however, we do freely make choices based upon our own dispositions, likings, and interests, which are largely shaped by nature and forces outside of our control. When Sabrina chooses to play the oboe, she freely does so because she has a gene that makes her like to play the oboe and nobody is making her do it. She could do something else, but she she’s choosing to play it because she likes it. Why does she like it? She has the “oboe gene” as it were.
I think this middle ground view is the right way to view the issue of free will. We all have our own predispositions and biases that are caused by forces outside of our control. We however have the freedom to act on these predispositions or not. These predispositions somewhat bind us, but they also allow us to experience free will at the same time. I hope that all made sense!