We began cohabiting in 2022, but I had my place at the time and didn't change my address on my documents. I moved in completely in 2023 and officially changed the address to my partner's. We have been living together for almost two years now. In my case, do I say I entered the common-law relationship in 2023 (a year after living together) or 2024 (a year after I officially changed my address)? I can still provide bills for expenses we shared in 2022, but my telephone/bank statements had my address.
As a general rule, I think you should assume that IRCC will be VERY skeptical about people claiming they were residing together full-time while maintaining two separate residences. This doesn't mean you can't convince them - with evidence - but 'part-time' (most of the time at one place) is not the same time as exclusively residing together. This is especially true for those whose 'other' residence is in a different city and esp province.
That's in general terms. You are vague about months. It would be BETTER if you have concrete, full documentary evidence you've lived together more than 12 months - so if, eg, it was February 2023, no problem. You can apply stating you began cohabitating in [month] 2022, but they need to know the PROVEN period is > 12 months.
This is important enough that if your best evidence is from March 2023, I'd wait and apply in four weeks. But this is dependent on how confident you are in the rest of your evidence (i.e. from the earlier period) such as bills and etc.
As far as the specific date issue - the instructions do say, if I remember correctly, that you are common law not from the date of beginning to cohabit but 12 months after the date of cohabitation. BUT: we have seen IRCC screw up and misunderstand this (although far more common is for applicants to make mistakes about the date and all).
If you have been living together for two years - and if you're close, consider waiting until it is two full years, I'd answer that question (therefore) as/at the date of the first anniversary. But make sure everywhere else, everywhere, you refer to the start of your cohabitation and emphasize that 'we have been cohabiting for two years beginning on [date].'