Illinois law requires the wedding ceremony be solemnized by someone who is legally authorized. If a priest married you, that would have been enough to validate your marriage. Then either a member of the clergy OR the two wedding participants can file the wedding certificate by registering it.
As to your ultimate goal, Illinois is a no-fault divorce state.
This means that the party who wants a divorce does not have to show a specific misbehavior on the part of the other spouse, such as adultery, abandonment, invalid marriage or the like. And, the method you proposed is not a likely avenue to gain an annulment.
The party simply has to state that there is no chance the marriage will be patched up, and that is sufficient grounds for a dissolution of marriage in Illinois.
This question never came up in law school. However, and based upon 47 years as a practicing lawyer, I would say that you are legally married.
The prior response is correct, however, the officiant must be licensed to solemnize the marriage and the marriage must also be registered. There are exceptions to this rule if both parties believe they were married, but in order to qualify for an exception, there is a very fact specific analysis that must take place. If certificate was filed and you were married by a licensed officiant, it would likely be difficult for you to invalidate the marriage.
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