A refusal usually has a reason, and the reason matters for what you do next.
“I don't believe in therapy.” Often it means not knowing what it looks like. One informational session, with no commitment, changes more than ten arguments.
“So I'm the problem.” Fear of being judged, or of ending up two against one. Worth saying explicitly: a couples therapist works with the dynamic, not with a culprit, and if a session turns into a tribunal, the therapist is making a methodological error.
“There's nothing to fix.” Sometimes the disagreement is precisely about whether a problem exists, and that is important information in itself.
What changes if you go alone: you learn what you do inside the repeating loop. It sounds like less than you wanted and it is more than it looks — most couple conflicts are self-sustaining, and if one side stops responding the same way, the pattern can no longer run identically. Sometimes that is what brings the other partner to a session, a few months later.
What cannot be done: couples therapy requires both partners. A therapist hearing only one version cannot work with the relationship — they can work with you inside the relationship, which is a different thing. For the same reason, your individual therapist is usually not the right one for later couples work: they have already heard one side.
One case where none of the above applies: in relationships involving violence, couples therapy is not indicated and can increase risk. There, the first step is safety and specialised individual support.