HELP PLEASE. Accidentally overstayed in Schengen

I (American citizen) am on a tier 4 UK student visa that expires 30 March 2020. Following my dissertation, I accompanied my fiancé (British citizen) to Sweden for a 3-month work placement that abruptly got extended to 5-months. What we didn't realize is that the rules for visiting as a tourist are very different in the UK than in the Schengen Zone; so when my fiancé's placement got extended, I was going to be overstaying the 90-day limit in Schengen and in our naïvity, we didn't consider that. I entered Schengen in early October 2019 via Ijmuiden in the Netherlands, went back to the UK via Calais over the holidays (after 84 days in Schengen), and then got granted re-entry via Copenhagen Airport on 4 January 2020.

I now find myself in Sweden having stupidly overstayed and I'm not sure what to do for fear of being blacklisted from entering the EU for the next decade or more. Will I be able to get back to the UK on my still valid student visa by car or plane? Or do I need to book a flight home to New York immediately? If caught, how likely is it that I’ll actually be blacklisted (considering that I haven’t been illegally employed, just merely existed in Sweden for too long)? The plan was originally to drive back to the Netherlands later this month and board the ferry back to Newcastle, but our research is saying that if I leave Schengen via the Netherlands, I'll get flagged up at that port as having entered in October (thus having overstayed by about 40 days).

I realize that there are risks involved no matter what we do and that the safest option is to avoid any more Schengen borders and just fly from Stockholm back to the United States ASAP. But optimally, I would like to avoid hastily returning to the US at this juncture, least of all because my fiancé would then be forced to drive all the way from Sweden to England alone with my very demanding cat who despises road trips. So, we are weighing the risks of changing our plans so that we instead drive to Belgium and take the ferry from Zeebruge to Hull, thereby avoiding my most recent Schengen points of entry/departure in the Netherlands and France. Has anyone driven between Denmark and Germany/Germany and Belgium in the last few days? Was border control in operation? We were able to get from Copenhagen to Brussels and then on to Calais in December with no checks, so that gives me hope. Basically, we are hoping that we'll be able to get back into the UK to sort out my tier 1 spousal visa and that no immigration officers will make a thorough study of my passport along the way (as my passport is full of a wide variety of EU stamps dating back to 2017 that would be time-consuming to decipher and passport control usually doesn't pay me much attention anyway).

I know any advice given on here comes with no guarantees, but any advice would be much appreciated nonetheless. Thank you.

You'll be so angry with yourself for not taking the safer option if you get caught.
And this comes from someone who likes to take risks.
Fly back to New York. Good luck.

Apply to extend your visit to Sweden

Contact your embassy immediately and explain the situation. If you show up at the airport having overstayed your visa by more than a couple of days they will take punitive measures and the fines for that are something stupid like 1200 euro and a potential travel ban. Explaining your situation at the airport isnt going to help you. Explaining your situation in advance might help your situation since It adds validity to your narrative of having made a whoopsie.

Trying to sneak through Schengen back to England might succeed but I highly advice you not to go ahead with that. If you get busted the situation will only be so much worse.

You can't extend a visa that is no longer valid.

That's the last thing you should do. That only gives them a reason to check. They'll go through everything from your past when they review your case and they'll be particularly interested in finding out whether or not you've overstayed.

What could my embassy possibly do though?

That's risky too. The person needs to be able to prove that they've made a mistake.

So you think I shouldn't contact the embassy and just book a flight to the US?

But why wouldn't it be safer to just fly back to the US?

What about flying to the UK or anywhere else outside of the Schengen Zone? I don't really have anywhere to go in the US; my parents haven't exactly kept a bedroom ready for me. My home is in England with my fiancé and my cat.

You failed to properly understand the rules. You failed to obey the rules. Why do you expect special treatment?