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Non Schengen citizens, such as those from the USA, are only allowed to stay in the Schengen zone聽 (such as Hungary, et al.) for only up to 90 days within a 180 day period as a tourist. You can not normally extend this tourist visa limit.

Have you been to the local Hungarian immigration office? I ask, since they will tell you pretty much what I said above. And that is the place to deal with this issue properly.

And you may find at that office, that practically speaking, you really only have 3 residency visa options, all of which are designed to make sure you will not become a social case on the state:

1) Non-work residency visa: you can not work here, but must show financial independence while living in Hungary. The visa to select if you are retired.

2) Work-visa: A local Hungarian business must sponsor you, and that is not easy (the business must show your work can not be done by a locally living Hungarian). If you loose your job, you have to find another employer sponsor, or leave the country.

3) Family reunification: Your wife asks the immigration office to give you a visa so you can live in Hungary with her. She then is the official "sponsor".

Normally, you should usually start any of the above from the USA, as you need to collect a lot of paperwork to process your application. And I doubt any visa application process now started will be completed in your remaining time frame. If you begin one of these processes, you may try asking for a special extension to remain until the visa application is processed (but I am only guessing about that as an idea -- you will need to ultimately ask about that at the immigration office)

For what it it worth, I do not think you need to have residency to get a marriage license in Hungary, and you can try asking for an extension on your stay until your marriage application goes through (but I am not 100% sure on that -- you really should ask someone in an official office about that as well). But, that is probably only a short term solution; I doubt you could extend that as long as until next summer.

But, quite frankly, I spent three years in a long term relationship with my wife, flying over "the pond" a great deal, before we finally tied the knot. Just saying, do not be in too much a hurry. A few months back in the States may be healthy; to help confirm the depth of the relationship.

I agree 1,000% with Klallee's answer.
My son married a HU women 15 years ago in Hungry but he is a dual citizen so no issue with overstaying.
It takes at least 30 some days after you apply to marry for the civil wedding to go through.
You must prove you are actually single and have all your birth papers etc. with you.
If anything is out of order then they will not go through with the wedding.
My son's current wife, his second wife is from Japan and she flew over 3 times and he went to Japan twice before they legally married in the US, they had a second wedding in Japan a year later.
Thing is, he never filed in Hungary to inform them he divorced his first Hungarian wife. I believe from what we have been told in HU in the city hall that he must file here in Hungary because that's where the wedding took place.
His ex wife from HU visited the US on a fiance visa but silly girl overstayed and they didn't marry on that trip. Made it impossible for her to return to the US for years unless they did get married. He traveled to HU because she overstayed in the US and couldn't come back until they were "legal" Even so she stayed in HU almost a full year after he left HU to be sure she got her legal papers to work in the US and all her US cards. Mess up and you pay.
If he ever steps foot in Hungary before getting things straighten out, I am afraid he will be arrested for bigamy , he must file with the marriage board here in Hungary. If you marry in Hungary then you have to follow HU rules even if you move away.
You may have to return to the US and then get a fiance visa, which my son did for both his former Hungarian wife and for his current Japanese wife. They give you 6 months or so to get married, if you don't do it, then you risk not being able to enter the country for several years time. At least that is how they do it in the US.
They take this seriously, in the US my son had to pay up the nose for paperwork and actually get documents from people stating they were actually a couple, sent in photos of their wedding to US immigration services etc. Hungary is just about as strict as the US is.
If you do marry in HU your wife will have to prove she can afford to keep you. if not enough funds in the banks etc. then sorry even if married you have to go home.
I went through allot of this in Hungary even though I have been married since 1978 to a HU citizen and our son is a citizen.
Sort of our fault because we tried doing the paperwork in Hungary before entering, entered in as a tourist.
Go into immigration and ask some questions while your still here and have a bit of time.