A desire to get out of the marriage in and of itself is NOT enough.
I’m not a theologian, so this is a layperson’s understanding. An annulment proceeding is a VERY different kind of proceeding than a divorce. For example, usually the spouses are NOT permitted to hear the testimony. Unlike common law, canon law assumes that witnesses are more likely to tell the truth if the spouses have no way of knowing what they said.
Usually, the evidence involves things that happened or existed PRIOR TO the marriage, not during it. So, the focus tends to be on whether there were any impediments to the marriage and whether the parties freely consented to the marriage.
For my generation, one of the most common grounds for an annulment was that the couple married because the woman was pregnant and they would not have married but for that. So, if you got engaged and after you agreed to marry, the wife became pregnant and you just moved up the ceremony that, in most cases, would NOT be grounds for an annulment. But if the couple had no intention of marrying and did so only because of the pregnancy, you’d probably get an annulment. (The Church implicitly recognizes it pushed a lot of people in this situation into marriage; it no longer does so.)
Another ground is that you’re “closely” related. The Church has MUCH stricter rules about marriage between relatives than most US states do. This is famously the ground upon which Rudy Guiliani had his first marriage annulled so he could marry Donna Hanover. His first wife was something like his third cousin, which is okay in New York State, but not the Church.
Another ground is not being “open to children.” I knew a devout Catholic who found out on her wedding night that her new H had had a vasectomy. Not only was he not “open to children,” but she didn’t give informed consent because she wouldn’t have married him if she had known.
In other situations, one or both spouses were not capable of consent or did not give informed consent.
Again, it’s usually what happened PRIOR to the marriage. If the marriage is declared invalid by the Church, then both parties are free to remarry in almost all cases. Again, fault is not considered.