I’m pretty sure that (at least in NC), separation requires different addresses, so your separation wouldn’t start until 2/1.
You can (when drafting the agreement, so I don’t know whether you can do it now), put in a requirement that the spouse retaining the house purchase the departing spouse’s interest and refinance the house in the resident spouse’s sole name. (Whether your spouse would be capable of complying is a different kettle of fish.)
It would be actively disrecommended to lie about the separation date. If your spouse wants a divorce faster, there are legal (but cumbersome) methods for doing so. (For example, one can take up legal residence in a state with a short residency requirement, and a minimal separation requirement, if any.)
From what I’ve been told, there are means of forcing a division of property (even after a divorce has been finalized), but if your house is already on the market, the proceedings may not do any good. See if you can get a referral to an attorney that does partitioning.