BETTER OFF implies there is some sort of choice. "I'll go to the store next week." "You're better off going today, etc. etc. YOU BETTER would be alright actually, though the correct phrase is YOU'D BETTER. Also, I just heard there WILL be a lockdown, not WOULD

You can say that but it is a different meaning than what they intend to say, what they mean is that it will be more appropriate and beneficial for you if you go to the store now.