You can apply now as common-law partners, IF you can prove you've been co-habitating for at least one continuous year. Read through Sections 5.34-5.35 of the
OP2 Processing Manual for information about how they assess common-law qualification. Otherwise, get married and apply as spouses.
You're out of status so, technically, you're in Canada illegally. That doesn't mean you won't be able to get a marriage licence but it does mean that, contrary to what you might interpret from information on the CIC website, applying for PR via the "inland" process is the worst thing you could do. First of all, there is no right of appeal for inland aps. Secondly, you'll be stuck in Canada for the duration of the process because if you leave and you're not re-admitted (likely because you've already overstayed) you lose the inland application and have to start all over again, and 3) you could wait for up to two years just to get to "first stage approval" where they issue you new temporary status - simply because of the fact that you're currently in Canada without valid status. For the inland process to go smoothly applicants have to have documented, valid temporary status in Canada and provide proof of that with their PR application. Yes, it is possible to apply and be approved for PR from within Canada even when you don't have valid status in Canada, but being in that situation means your application cannot benefit from straight-forward processing and it will be transferred to your local office for processing. Local offices are often backlogged with applications (especially in areas with a high influx of immigrants) and it can take up to two years for an officer to even start to review your file.
So - it is possible to apply for PR via the
outland application process even while you're in Canada. It won't get you new temporary status - there's no way for you to do that now except to leave the country and try to re-enter . . . understanding that they can always refuse to allow you back in. But an outland PR application will continue to process without issue, regardless of where you're staying. If you left Canada and couldn't get back in, it would continue to process. If you stay in Canada illegally, it will still process. That's because it's processed through the embassy that represents your country of citizenship (Buffalo in your case), irregardless of where you're staying/living at the time. And it's a faster process than the inland one - Buffalo is currently finalizing spousal/common-law applications in 4-10 months. Add to that another 60 days or so for the sponsor's assessment through the Case Processing Centre in Mississauga (which happens before the file is transferred to Buffalo), and another 6 weeks if you're in Quebec. If an interview is required (less likely for spousal applicants because you don't have the additional burden of proof of common-law status), you'll have to go to the States for it and you might not get back into Canada afterwards BUT it's still a faster process AND you maintain your right of appeal for a refusal.
Either way - whether you get married or you apply as common-law partners - do it ASAP. Check out the
US2Canada website for more specific information about the requirements.