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It is rather unlikely that the US Embassy would deny you a passport renewal (although some countries do just that), you should just try. Reporting you to the Bulgarian authorities is even more unlikely, and you would sooner or later be detected anyway (at the border, when leaving Bulgaria, for example.

As to entering the US... you, being a citizen, can go there without vaccination, you will need a test though. Hope you don't religiously object to that. Your wife, needing a visa, is a different matter  She may or may not be able to enter.

But be assured that in the worst case you can get her a Green Card in a few months, and you can live in the States.

You're still an American citizen so I don't believe the US embassy would really care that you over stayed your visa, that doesn't affect your citizenship and they are required to help any American citizen.  I don't know if they are required to report anything to the Bulgarian government though so you might just want to start off anonymously and see what they say.  I've spoken to them multiple times since I'm moving there next year and they have always been extremely polite and helpful.  They should be able to help you with your passport.

I found this on the Department of State website that specifically addresses returning to the US,

It's a pickle indeed.

You're lucky that Bulgaria takes such a low-key approach to illegal immigrants!

Is your wife doing all the driving? Or are you driving illegally too?

You should see an immigration lawyer and get advice on how to regularize your situation ASAP.

However, I doubt that the US embassy will do any checking on your immigration status in Bulgaria. So it should be no problem to renew it.

If you don't want to be separated... maybe you can both sneak over the border to Romania by car, so you're out of the country, and nobody knows you overstayed.  Then you hang out in Bucharest for a while, and use your new passport to apply for your D visa there (on the basis of your marriage), then come back in again legally?

I very much doubt that the US embassy is concerned with your Bulgarian visa issues. However, what I personally would be concerned about is that when you leave a country after a sizeable overstay you are quite often given a deportation stamp which makes it practically impossible to ever renter that country again should you wish to. I’m only saying this from meeting the odd hapless foreigners finding themselves in similar situations to yourself whilst I was living abroad for many years in non EU countries. It could not be the case with Bulgaria but you have basically taken a massive liberty with their immigration law.
I would get an immigration lawyer  ASAP and see if they can rectify the overstay issue. I’m thinking (perhaps erroneously) that a little cash may help in this matter

Adminstrative fines in Bulgaria are pretty low. For example, not filling in your census form is 160 leva. So the overstaying fine is probably not outrageous (but the potential bar on re-rentry would be a problem).

The other good news is Bulgaria is in the EU, and has land borders. If you can only fly out of a country, then it's hard to avoid getting caught.

We accidentally overstayed in our house in Germany a couple of times (my partner is non-EU), but we just drove to France, and all was fine. To be on the safe side, I bought a couple of extra ferry tickets to "prove" that our last visit was only a few weeks rather than a few months.

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Evtra makes a good point, if you are legally married to a Bulgarian citizen, that means you wouldn't have had to go though all the hoop-jumping other non-EU citizens do to get a D visa and if you'd applied straight away wouldn't be in this pickle now.
Is this a marriage that the state of Bulgaria would legally recognise, or was it a personal or religious ceremony they probably would not?
You also need to consider that with only two months left on your existing passport, you may have problems getting a visa to enter another country apart from the US, as most will want at least six months validity to give you a visa or allow visaless entry for those countries that don't need a visa.
Adding to the chorus here - you've gotten yourself in a mess, left it to the last minute to try to sort it out, and need to get good legal advice ASAP, and, as well, start the process of applying for a new passport immediately. Let that passport expire and you're in a REAL mess!