Upvote:3
After the marriage, you would be in the Immediate Relative category (spouse, parent, or unmarried under-21 child of a US citizen). Immediate Relatives are eligible for Adjustment of Status in the US even if they entered on the Visa Waiver Program (whereas other categories cannot do AOS from VWP). So yes, you can stay and apply for Adjustment of Status. If you do Adjustment of Status, you cannot leave the US until you get your green card or an Advance Parole, or you will automatically abandon your AOS. The only issue with AOS from VWP is that people who entered on VWP waived any right to a hearing before being deported (except when claiming asylum), so in the unlikely case where the AOS is denied in error, you don't get to challenge the denial in removal proceedings.
Upvote:14
Knowing someone who did exactly this in the US very recently, I can say that it is possible to change your status once you are married without having to leave. However the law around it is not clear or simple and you should absolutely definitely talk to an immigration lawyer before attempting this. It's very easy to do something, or fail to do something, that would mean you didn't qualify and were overstaying your VWP limits, causing you lots of problems in the future.
My understanding is that you not only need to be married, but also to have filed the paperwork for the status change, before your VWP period expires. But your lawyer will know, so listen to them not me.
Do not ask a city hall official. There is no reason why they would know immigration law.