If the property goes back to the bank the tenant will have to leave (by choice or force). If a 3rd party investor picks up the property the tenant could try to negotiate a new lease but it's ultimately the decision of that new owner.
It is interesting to note that some courts across the nation are halting all eviction proceedings of tenant-occupied properties after foreclosure. Great for the tenants since they were most likely keeping their end of the lease in good standing but not so good for banks or investors who don't wish to be landlords.
Regardless, the tenants lose when the property is foreclosed. Security deposit... forget about it! Your only recourse is to sue the landlord but I doubt that will get you anywhere.
But, judging from the last tax roll for the property address you referenced, she does have a mortgage, and the bank will rarely ever let this happen, because they become a junior to county taxes, and they risk losing their interest in the property over a few thousand dollars. So even if the land lady was not escrowing their taxes, the mortgage companies have a whole department who just go and make sure taxes are paid on the homes they have liens for.
So if in fact in the property you are living in was just good "ole' fashioned foreclosed" on yes there is a possibility you can do a rent back, but keep in mind you will probably not have a good lease, and your property may be on the market, meaning people going through your home who are interested buyers. If you are not getting any straight answers from your land lady, I would suggest calling, or going to the Denton County Tax Office and inquire with them, they will have record if your house has been foreclosed on too if it was done recorded a month or 2 ago. Also, if you are in a financial position, and are interested, now is the to purchase it, thank bank would love to not have to mess with it anymore either.