The official site lists the requirements for Arbeitslosengeld:
You were employed and payed into the insurance for at least 12 of the last 30 months. ✓
You notified the Amt well in advance (for limited contracts that expire it's 3 months I think) ✓
You are actively cooperating with the Amt and looking for employment. ✓
You are unemployed but could take up employment
Now, the last one is tricky. Technically, you don't have a work permit when your contract runs out. You could get one if you got a job. Seems like a catch-22 that you need an employment lawyer to answer.
If you were to marry a German citizen and subsequently apply for Familienzusammenführung ("family reunion"), you will be granted first a temporary residence permit for 3 years and if you don't commit any major crimes in that time, you will be granted a permanent residence permit thereafter. Residence permits by default include unrestricted working permits. Since you are a US citizen, you do not need to go back to your home country and apply from the embassy there, you can do it by just walking in the German Ausländerbehörde here. Please note that "walking in" was maybe kind of an euphemism from ten years ago, you need to make an appointment nowadays. You will need a lot of authorized, notarized, apostilled, certified translated documents to marry and German Amts are pretty strict in that regard. If you don't have those documents yet, go get them. Three months is a tight timetable for this. The US operates on the principle of "we trust you, you can marry, we will punish you later if we find out made false claims" but Germany is one of those "we don't trust you, prove everything you just said or we won't let you marry" countries. Prepare for paperwork.
That said, me not being a lawyer, you should be able to get Arbeitslosengeld when you are married and then got your permanent residence permit. You might need a lawyer to get it though, since people working at Amts are really not good at fringe edge cases like yours will be and they are not lawyers either. You being right does not automatically mean the person that gets your application agrees with you without a lawyer paid by you telling them that you are in fact correct. Again... prepare for paperwork. I think that should be our state motto. Germany - "Prepare for paperwork".
So to summarize:
Whether you are eligible for unemployment benefits depends on your work permit after your contract ends. I could see a case in your favor since getting a job would mean getting a permit, but I could also see it through the eyes of buerocracy, where if you don't have a permit to copy and file right now, you won't get benefits. This is lawyer territory.
Marrying would solve that problem, but it is complicated in Germany. At least compared to the US. And after that you still need to apply for family reunion and while that is guaranteed to be granted in your case, it will take time, too. So if you aren't very very good at interacting with the German buerocracy or have lead a very, very simple life up to now that needs very little paperwork, three months might simply not be enough time to actually do all that.
As far as the job seeker visa goes... to be honest I don't know and I could not find anything about it. I would assume it's the same fringe edge case I talked about before where technically you might be correct about the law, but I would expect the Amt to deny you at first. A job seeker visa requires you to show enough funds to support yourself, so I don't think it's meant to be used to get money from the government.
