As I say, the laws vary by jurisidiction. But it's unlikely (heck, it's impossible) that you could just stay in the house, keep paying your rent every month, and ignore the landlord's request that you sign a lease.
If there had previously been a lease agreement, the new landlord would have been obligated to observe the terms and conditions of the existing lease. But since there wasn't a lease agreement, there's no previous lease he has to obey. And it's certainly within his rights to ask you to sign a lease.
Check with a lawyer. But if the landlord wants you to sign a lease and you decline to do so, it's very likely that the landlord can require you to move.
Hope that helps.