Not renewing your lease isn't the same as evicting you....They might increase your rent if there's a month to month fee...
Have you reclocated the dog to a friends house in the mean time?
Just keep the manager informed that you plan to move etc...Removing the dog would be a good thing
FYI - In Texas - Evictions
1) They'll give ya a three day notice to vacate
2) On day four if you're still there they'll file a formal eviction with the court
3) You'll get served paper to appear in court in 7 to 10 days
4) Both parties will present their case before the judge
5) Judge will make his/her rulling - will probably give you ten days or so to get out (provided you lose)
6) If you don't get out when ordered - the manager can physically toss everything you own on a public street....