If I was unable to give a 30 day notice before moving, doI have to pay the next month’s rent?

If you did not give the required notice before moving, you remain liable for the rent until the place has been re-rented or for the balance of the term of your lease.  The landlord cannot allow the place to remain vacant for the balance of the term of your lease and must make reasonable efforts to re-rent the place.  If the landlord fails to make reasonable efforts to find another tenant, the landlord has failed to mitigate (minimize) damages and the landlord's damages will be reduced accordingly.  Damages means the amount of compensation the landlord is seeking to recover from you in a lawsuit.  Reasonable efforts to attract another tenant would be determined by what other landlords in the area are doing such as running an ad in the newspaper, posting a vacancy sign on the property, advertising in local rental guides, etc.

If all that is required is a thirty day notice, you should give the notice immediately so that you can end your obligation to pay rent thirty days from now.  If you don't give any notice, you are liable for the rent for the balance of the term of your lease or until the place is re-rented whichever occurs first as mentioned above.