I can't tell from your question whether you were still under your signed lease at the time or were renting on a month to month basis. Your lease will govern whether you are obligated to pay the extra month. In addition, the Texas Property Code has provisions regarding month to month leases.
The collection agency and their activities are a separate issue in that there are rules regulating their activities, even when the debt is legitimate and owed.
As a practical matter, it may be possible (if you in fact owe the money) to simply work out a payment agreement with your previous landlord. That could stop the collection activities, preserve your credit and hopefully keep your former landlord as a positive reference in the future.