Legally, you are month to month until you sign the new lease on the terms of your original lease. An email is not a binding contract.

However, it’s a tricky situation. Because you are month to month, if you decide to not pay the updated rent the landlord can decide to not send you the new lease and instead give you notice to leave(60 days if you’ve been there 1-2 years, 90 days for longer).

So basically you have to decide if that risk is worth it. Either way I would definitely advise you remind them they never actually sent you the new lease . You can try saying you expect the new lease will take effect February first and see what they say.